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{
"T237": {
"Title": "Characteristic polynomials of connected graphs",
"Numbers": {
"A_": {
"number": "x^2 - 1",
"comment": "This is the complete graph $K_2$."
},
"BW": {
"number": "x^3 - 2*x",
"comment": "This is the path $P_3$."
},
"Bw": {
"number": "x^3 - 3*x - 2",
"comment": "This is the complete graph $K_3$."
},
"CF": {
"number": "x^4 - 3*x^2",
"comment": "This is the star $K_{1,3}$."
},
"CL": {
"number": "x^4 - 3*x^2 + 1",
"comment": "This is the path $P_4$."
},
"CN": "x^4 - 4*x^2 - 2*x + 1",
"C]": {
"number": "x^4 - 4*x^2",
"comment": "This is the cycle $C_4$."
},
"C^": {
"number": "x^4 - 5*x^2 - 4*x",
"comment": "This is the diamond graph."
},
"C~": {
"number": "x^4 - 6*x^2 - 8*x - 3",
"comment": "This is the complete graph $K_4$."
},
"D?{": {
"number": "x^5 - 4*x^3",
"comment": "This is the star $K_{1,4}$."
},
"D@s": "x^5 - 4*x^3 + 2*x",
"D@{": "x^5 - 5*x^3 - 2*x^2 + 2*x",
"DIk": "x^5 - 5*x^3 + 2*x",
"DBk": {
"number": "x^5 - 5*x^3 - 2*x^2 + 3*x",
"comment": "This is the bull graph."
},
"DB{": {
"number": "x^5 - 6*x^3 - 4*x^2 + 2*x",
"comment": "This is the dart graph."
},
"DFw": {
"number": "x^5 - 6*x^3",
"comment": "This is the complete bipartite graph $K_{2,3}$."
},
"DF{": "x^5 - 7*x^3 - 6*x^2",
"DBg": {
"number": "x^5 - 4*x^3 + 3*x",
"comment": "This is the path $P_5$."
},
"DK[": "x^5 - 5*x^3 - 2*x^2 + 4*x + 2",
"DK{": {
"number": "x^5 - 6*x^3 - 4*x^2 + 5*x + 4",
"comment": "This is the butterfly graph."
},
"DLo": {
"number": "x^5 - 5*x^3 + 5*x - 2",
"comment": "This is the cycle $C_5$."
},
"Dbk": {
"number": "x^5 - 6*x^3 - 2*x^2 + 4*x",
"comment": "This is the house graph."
},
"DL{": "x^5 - 7*x^3 - 6*x^2 + 3*x + 2",
"DJk": "x^5 - 6*x^3 - 4*x^2 + 3*x + 2",
"DJ{": "x^5 - 7*x^3 - 8*x^2 + 2",
"DN{": {
"number": "x^5 - 8*x^3 - 10*x^2 - x + 2",
"comment": "This is the house X graph."
},
"DNw": "x^5 - 7*x^3 - 4*x^2 + 2*x",
"D]{": {
"number": "x^5 - 8*x^3 - 8*x^2",
"comment": "This is the wheel $W_5$."
},
"D^{": "x^5 - 9*x^3 - 14*x^2 - 6*x",
"D~{": {
"number": "x^5 - 10*x^3 - 20*x^2 - 15*x - 4",
"comment": "This is the complete graph $K_5$."
},
"E?Bw": {
"number": "x^6 - 5*x^4",
"comment": "This is the star $K_{1,5}$."
},
"E?Fg": "x^6 - 5*x^4 + 3*x^2",
"E?Fw": "x^6 - 6*x^4 - 2*x^3 + 3*x^2",
"E@FG": "x^6 - 5*x^4 + 5*x^2",
"E?NG": "x^6 - 5*x^4 + 4*x^2",
"E@JW": "x^6 - 6*x^4 + 4*x^2",
"E?NW": "x^6 - 6*x^4 - 2*x^3 + 5*x^2",
"E?Nw": "x^6 - 7*x^4 - 4*x^3 + 4*x^2",
"E?]o": "x^6 - 6*x^4 + 5*x^2",
"E?^o": "x^6 - 7*x^4 + 3*x^2",
"E?]w": "x^6 - 7*x^4 - 4*x^3 + 5*x^2",
"E?^w": "x^6 - 8*x^4 - 6*x^3 + 3*x^2",
"E?~o": {
"number": "x^6 - 8*x^4",
"comment": "This is the complete bipartite graph $K_{2,4}$."
},
"E?~w": "x^6 - 9*x^4 - 8*x^3",
"E@QW": "x^6 - 5*x^4 + 5*x^2 - 1",
"E@`w": "x^6 - 6*x^4 - 2*x^3 + 6*x^2 + 2*x - 1",
"E@Rw": {
"number": "x^6 - 7*x^4 - 4*x^3 + 7*x^2 + 4*x - 1",
"comment": "It has the same characteristic polynomial as `E@]o`."
},
"E@YW": "x^6 - 6*x^4 + 5*x^2 - 1",
"EHQW": "x^6 - 6*x^4 + 8*x^2 - 2*x - 1",
"E@UW": "x^6 - 6*x^4 - 2*x^3 + 6*x^2 - 1",
"EGNW": "x^6 - 7*x^4 - 2*x^3 + 7*x^2 - 1",
"EAMw": "x^6 - 7*x^4 - 4*x^3 + 6*x^2 + 2*x - 1",
"E@Vw": "x^6 - 8*x^4 - 6*x^3 + 6*x^2 + 2*x - 1",
"E@YO": {
"number": "x^6 - 5*x^4 + 6*x^2 - 1",
"comment": "This is the path $P_6$."
},
"EI_w": "x^6 - 6*x^4 + 6*x^2",
"EGdo": "x^6 - 6*x^4 - 2*x^3 + 8*x^2 + 4*x - 1",
"EGcw": "x^6 - 6*x^4 - 2*x^3 + 7*x^2 + 2*x - 1",
"E@ow": "x^6 - 6*x^4 - 2*x^3 + 7*x^2 + 4*x",
"EK`w": "x^6 - 7*x^4 - 2*x^3 + 8*x^2 + 4*x",
"ECXw": "x^6 - 7*x^4 - 4*x^3 + 7*x^2 + 4*x",
"EGdw": "x^6 - 7*x^4 - 4*x^3 + 9*x^2 + 6*x - 1",
"E_Nw": "x^6 - 8*x^4 - 6*x^3 + 9*x^2 + 8*x",
"E@NW": "x^6 - 7*x^4 - 4*x^3 + 6*x^2 + 4*x",
"E@Nw": "x^6 - 8*x^4 - 8*x^3 + 3*x^2 + 4*x",
"E_]o": "x^6 - 7*x^4 + 9*x^2 - 4*x",
"EENg": "x^6 - 8*x^4 - 2*x^3 + 7*x^2",
"EC\\w": "x^6 - 8*x^4 - 6*x^3 + 5*x^2 + 4*x",
"E@^W": "x^6 - 8*x^4 - 6*x^3 + 7*x^2 + 4*x - 1",
"E_]w": "x^6 - 8*x^4 - 4*x^3 + 8*x^2",
"E@vw": "x^6 - 9*x^4 - 8*x^3 + 6*x^2 + 4*x",
"E@^o": "x^6 - 8*x^4 - 4*x^3 + 6*x^2 + 2*x - 1",
"E@]w": "x^6 - 8*x^4 - 8*x^3 + 4*x^2 + 4*x - 1",
"E@^w": "x^6 - 9*x^4 - 10*x^3 + 3*x^2 + 4*x - 1",
"E@~o": "x^6 - 9*x^4 - 4*x^3 + 4*x^2",
"E@~w": "x^6 - 10*x^4 - 12*x^3 + x^2 + 4*x",
"EBYW": "x^6 - 7*x^4 + 4*x^2",
"EBYw": "x^6 - 8*x^4 - 4*x^3 + 6*x^2",
"EBZw": "x^6 - 9*x^4 - 8*x^3 + 4*x^2",
"EIe_": {
"number": "x^6 - 6*x^4 + 9*x^2 - 4",
"comment": "This is the cycle $C_6$."
},
"EKNG": "x^6 - 7*x^4 + 7*x^2 - 1",
"EHUW": "x^6 - 7*x^4 - 2*x^3 + 8*x^2",
"EoLW": "x^6 - 7*x^4 - 2*x^3 + 11*x^2 + 2*x - 4",
"E@]o": {
"number": "x^6 - 7*x^4 - 4*x^3 + 7*x^2 + 4*x - 1",
"comment": "It has the same characteristic polynomial as `E@Rw`."
},
"EA]o": "x^6 - 7*x^4 - 2*x^3 + 8*x^2 + 2*x - 1",
"EaMw": "x^6 - 8*x^4 - 4*x^3 + 9*x^2 + 4*x - 1",
"EBhw": "x^6 - 8*x^4 - 6*x^3 + 8*x^2 + 6*x",
"EBjW": "x^6 - 8*x^4 - 4*x^3 + 11*x^2 + 4*x - 4",
"EBjw": "x^6 - 9*x^4 - 8*x^3 + 9*x^2 + 8*x - 1",
"EImo": "x^6 - 8*x^4 + 4*x^2",
"EJYW": "x^6 - 8*x^4 - 4*x^3 + 7*x^2 + 4*x",
"EIno": "x^6 - 9*x^4 - 4*x^3 + 7*x^2",
"EImw": "x^6 - 9*x^4 - 6*x^3 + 6*x^2 + 4*x",
"EInw": "x^6 - 10*x^4 - 10*x^3 + 5*x^2 + 4*x",
"EI]w": "x^6 - 9*x^4 - 8*x^3 + 5*x^2 + 4*x",
"EB]w": "x^6 - 9*x^4 - 10*x^3 + 4*x^2 + 6*x",
"EB^w": "x^6 - 10*x^4 - 14*x^3 - x^2 + 4*x",
"EHuw": "x^6 - 9*x^4 - 6*x^3 + 8*x^2 + 2*x - 1",
"EBnW": "x^6 - 9*x^4 - 8*x^3 + 9*x^2 + 6*x - 4",
"EBnw": "x^6 - 10*x^4 - 12*x^3 + 4*x^2 + 6*x - 1",
"EB~w": "x^6 - 11*x^4 - 16*x^3 - 2*x^2 + 4*x",
"EFz_": {
"number": "x^6 - 9*x^4",
"comment": "This is the complete bipartite graph $K_{3,3}$."
},
"Es\\w": "x^6 - 10*x^4 - 6*x^3 + 3*x^2",
"EB~o": "x^6 - 10*x^4 - 8*x^3 + 4*x^2",
"EFzw": "x^6 - 11*x^4 - 12*x^3",
"EF~w": "x^6 - 12*x^4 - 20*x^3 - 9*x^2",
"EPTW": "x^6 - 7*x^4 - 4*x^3 + 8*x^2 + 6*x",
"E`NG": "x^6 - 7*x^4 - 4*x^3 + 11*x^2 + 12*x + 3",
"E`NW": "x^6 - 8*x^4 - 6*x^3 + 11*x^2 + 14*x + 4",
"E`Lw": "x^6 - 8*x^4 - 8*x^3 + 6*x^2 + 10*x + 3",
"E`Nw": "x^6 - 9*x^4 - 10*x^3 + 9*x^2 + 18*x + 7",
"EJeg": "x^6 - 8*x^4 - 4*x^3 + 12*x^2 + 8*x",
"EJqw": "x^6 - 9*x^4 - 6*x^3 + 11*x^2 + 8*x - 1",
"EK\\w": "x^6 - 9*x^4 - 10*x^3 + 5*x^2 + 10*x + 3",
"EK]w": "x^6 - 9*x^4 - 8*x^3 + 10*x^2 + 12*x + 3",
"E`]w": "x^6 - 9*x^4 - 8*x^3 + 8*x^2 + 8*x",
"EK^w": "x^6 - 10*x^4 - 12*x^3 + 7*x^2 + 14*x + 4",
"EK~o": "x^6 - 10*x^4 - 8*x^3 + 9*x^2 + 8*x",
"EK~w": "x^6 - 11*x^4 - 16*x^3 + 3*x^2 + 16*x + 7",
"EBn_": "x^6 - 8*x^4 - 2*x^3 + 10*x^2 - 2*x - 1",
"E`]o": "x^6 - 8*x^4 - 4*x^3 + 12*x^2 + 4*x - 5",
"ELpw": "x^6 - 9*x^4 - 6*x^3 + 11*x^2 + 4*x - 4",
"ELrw": {
"number": "x^6 - 10*x^4 - 10*x^3 + 10*x^2 + 8*x - 5",
"comment": "This is the wheel $W_6$."
},
"ELv_": "x^6 - 9*x^4 - 4*x^3 + 12*x^2",
"Ek]w": "x^6 - 10*x^4 - 8*x^3 + 9*x^2 + 4*x - 1",
"EJnW": "x^6 - 10*x^4 - 10*x^3 + 8*x^2 + 8*x",
"Ebnw": "x^6 - 11*x^4 - 14*x^3 + 4*x^2 + 8*x",
"EL~w": "x^6 - 12*x^4 - 20*x^3 - 4*x^2 + 8*x + 3",
"ER^W": "x^6 - 10*x^4 - 10*x^3 + 6*x^2 + 6*x - 1",
"EJmw": "x^6 - 10*x^4 - 12*x^3 + 5*x^2 + 12*x + 4",
"EJnw": "x^6 - 11*x^4 - 16*x^3 + x^2 + 10*x + 3",
"EJ]w": "x^6 - 10*x^4 - 14*x^3 + 8*x + 3",
"EJ^w": "x^6 - 11*x^4 - 20*x^3 - 9*x^2 + 4*x + 3",
"EJ~w": "x^6 - 12*x^4 - 22*x^3 - 9*x^2 + 6*x + 4",
"EN~w": "x^6 - 13*x^4 - 26*x^3 - 15*x^2 + 2*x + 3",
"EL~o": "x^6 - 11*x^4 - 12*x^3 + 5*x^2 + 4*x",
"EJ~o": "x^6 - 11*x^4 - 14*x^3 + 4*x",
"Ejmw": "x^6 - 11*x^4 - 12*x^3 + 3*x^2 + 4*x - 1",
"ENzw": "x^6 - 12*x^4 - 18*x^3 - 3*x^2 + 4*x",
"E]~o": "x^6 - 12*x^4 - 16*x^3",
"E]~w": "x^6 - 13*x^4 - 24*x^3 - 12*x^2",
"E^~w": "x^6 - 14*x^4 - 32*x^3 - 27*x^2 - 8*x",
"E~~w": {
"number": "x^6 - 15*x^4 - 40*x^3 - 45*x^2 - 24*x - 5",
"comment": "This is the complete graph $K_6$."
}
}
},
"T238": {
"Title": "Laplacian polynomials of connected graphs",
"Numbers": {
"A_": {
"number": "x^2 - 2*x",
"comment": "This is the complete graph $K_2$."
},
"BW": {
"number": "x^3 - 4*x^2 + 3*x",
"comment": "This is the path $P_3$."
},
"Bw": {
"number": "x^3 - 6*x^2 + 9*x",
"comment": "This is the complete graph $K_3$."
},
"CF": {
"number": "x^4 - 6*x^3 + 9*x^2 - 4*x",
"comment": "This is the star $K_{1,3}$."
},
"CL": {
"number": "x^4 - 6*x^3 + 10*x^2 - 4*x",
"comment": "This is the path $P_4$."
},
"CN": "x^4 - 8*x^3 + 19*x^2 - 12*x",
"C]": {
"number": "x^4 - 8*x^3 + 20*x^2 - 16*x",
"comment": "This is the cycle $C_4$."
},
"C^": {
"number": "x^4 - 10*x^3 + 32*x^2 - 32*x",
"comment": "This is the diamond graph."
},
"C~": {
"number": "x^4 - 12*x^3 + 48*x^2 - 64*x",
"comment": "This is the complete graph $K_4$."
},
"D?{": {
"number": "x^5 - 8*x^4 + 18*x^3 - 16*x^2 + 5*x",
"comment": "This is the star $K_{1,4}$."
},
"D@s": "x^5 - 8*x^4 + 20*x^3 - 18*x^2 + 5*x",
"D@{": "x^5 - 10*x^4 + 32*x^3 - 38*x^2 + 15*x",
"DIk": "x^5 - 10*x^4 + 34*x^3 - 46*x^2 + 20*x",
"DBk": {
"number": "x^5 - 10*x^4 + 33*x^3 - 40*x^2 + 15*x",
"comment": "This is the bull graph."
},
"DB{": {
"number": "x^5 - 12*x^4 + 49*x^3 - 78*x^2 + 40*x",
"comment": "This is the dart graph."
},
"DFw": {
"number": "x^5 - 12*x^4 + 51*x^3 - 92*x^2 + 60*x",
"comment": "This is the complete bipartite graph $K_{2,3}$."
},
"DF{": "x^5 - 14*x^4 + 69*x^3 - 140*x^2 + 100*x",
"DBg": {
"number": "x^5 - 8*x^4 + 21*x^3 - 20*x^2 + 5*x",
"comment": "This is the path $P_5$."
},
"DK[": "x^5 - 10*x^4 + 34*x^3 - 44*x^2 + 15*x",
"DK{": {
"number": "x^5 - 12*x^4 + 50*x^3 - 84*x^2 + 45*x",
"comment": "This is the butterfly graph."
},
"DLo": {
"number": "x^5 - 10*x^4 + 35*x^3 - 50*x^2 + 25*x",
"comment": "This is the cycle $C_5$."
},
"Dbk": {
"number": "x^5 - 12*x^4 + 51*x^3 - 90*x^2 + 55*x",
"comment": "This is the house graph."
},
"DL{": "x^5 - 14*x^4 + 70*x^3 - 146*x^2 + 105*x",
"DJk": "x^5 - 12*x^4 + 50*x^3 - 82*x^2 + 40*x",
"DJ{": "x^5 - 14*x^4 + 69*x^3 - 136*x^2 + 80*x",
"DN{": {
"number": "x^5 - 16*x^4 + 93*x^3 - 230*x^2 + 200*x",
"comment": "This is the house X graph."
},
"DNw": "x^5 - 14*x^4 + 71*x^3 - 154*x^2 + 120*x",
"D]{": {
"number": "x^5 - 16*x^4 + 94*x^3 - 240*x^2 + 225*x",
"comment": "This is the wheel $W_5$."
},
"D^{": "x^5 - 18*x^4 + 120*x^3 - 350*x^2 + 375*x",
"D~{": {
"number": "x^5 - 20*x^4 + 150*x^3 - 500*x^2 + 625*x",
"comment": "This is the complete graph $K_5$."
},
"E?Bw": {
"number": "x^6 - 10*x^5 + 30*x^4 - 40*x^3 + 25*x^2 - 6*x",
"comment": "This is the star $K_{1,5}$."
},
"E?Fg": "x^6 - 10*x^5 + 33*x^4 - 46*x^3 + 28*x^2 - 6*x",
"E?Fw": "x^6 - 12*x^5 + 48*x^4 - 82*x^3 + 63*x^2 - 18*x",
"E@FG": "x^6 - 10*x^5 + 35*x^4 - 52*x^3 + 32*x^2 - 6*x",
"E?NG": "x^6 - 10*x^5 + 34*x^4 - 48*x^3 + 29*x^2 - 6*x",
"E@JW": "x^6 - 12*x^5 + 51*x^4 - 96*x^3 + 80*x^2 - 24*x",
"E?NW": "x^6 - 12*x^5 + 50*x^4 - 88*x^3 + 67*x^2 - 18*x",
"E?Nw": "x^6 - 14*x^5 + 69*x^4 - 148*x^3 + 140*x^2 - 48*x",
"E?]o": "x^6 - 12*x^5 + 52*x^4 - 100*x^3 + 84*x^2 - 24*x",
"E?^o": "x^6 - 14*x^5 + 72*x^4 - 170*x^3 + 184*x^2 - 72*x",
"E?]w": "x^6 - 14*x^5 + 70*x^4 - 152*x^3 + 144*x^2 - 48*x",
"E?^w": "x^6 - 16*x^5 + 93*x^4 - 242*x^3 + 284*x^2 - 120*x",
"E?~o": {
"number": "x^6 - 16*x^5 + 96*x^4 - 272*x^3 + 368*x^2 - 192*x",
"comment": "This is the complete bipartite graph $K_{2,4}$."
},
"E?~w": "x^6 - 18*x^5 + 120*x^4 - 368*x^3 + 528*x^2 - 288*x",
"E@QW": "x^6 - 10*x^5 + 35*x^4 - 52*x^3 + 31*x^2 - 6*x",
"E@`w": "x^6 - 12*x^5 + 51*x^4 - 94*x^3 + 72*x^2 - 18*x",
"E@Rw": "x^6 - 14*x^5 + 70*x^4 - 156*x^3 + 153*x^2 - 54*x",
"E@YW": "x^6 - 12*x^5 + 52*x^4 - 100*x^3 + 83*x^2 - 24*x",
"EHQW": "x^6 - 12*x^5 + 53*x^4 - 106*x^3 + 95*x^2 - 30*x",
"E@UW": "x^6 - 12*x^5 + 51*x^4 - 92*x^3 + 69*x^2 - 18*x",
"EGNW": "x^6 - 14*x^5 + 72*x^4 - 168*x^3 + 176*x^2 - 66*x",
"EAMw": "x^6 - 14*x^5 + 71*x^4 - 158*x^3 + 149*x^2 - 48*x",
"E@Vw": "x^6 - 16*x^5 + 94*x^4 - 250*x^3 + 297*x^2 - 126*x",
"E@YO": {
"number": "x^6 - 10*x^5 + 36*x^4 - 56*x^3 + 35*x^2 - 6*x",
"comment": "This is the path $P_6$."
},
"EI_w": "x^6 - 12*x^5 + 53*x^4 - 106*x^3 + 92*x^2 - 24*x",
"EGdo": "x^6 - 12*x^5 + 53*x^4 - 104*x^3 + 84*x^2 - 18*x",
"EGcw": "x^6 - 12*x^5 + 52*x^4 - 98*x^3 + 76*x^2 - 18*x",
"E@ow": "x^6 - 12*x^5 + 52*x^4 - 98*x^3 + 75*x^2 - 18*x",
"EK`w": {
"number": "x^6 - 14*x^5 + 73*x^4 - 176*x^3 + 192*x^2 - 72*x",
"comment": "It has the same Laplacian polynomial as EBYW."
},
"ECXw": "x^6 - 14*x^5 + 72*x^4 - 166*x^3 + 164*x^2 - 48*x",
"EGdw": "x^6 - 14*x^5 + 72*x^4 - 166*x^3 + 165*x^2 - 54*x",
"E_Nw": "x^6 - 16*x^5 + 95*x^4 - 260*x^3 + 324*x^2 - 144*x",
"E@NW": "x^6 - 14*x^5 + 71*x^4 - 158*x^3 + 148*x^2 - 48*x",
"E@Nw": "x^6 - 16*x^5 + 93*x^4 - 238*x^3 + 256*x^2 - 96*x",
"E_]o": "x^6 - 14*x^5 + 74*x^4 - 184*x^3 + 216*x^2 - 96*x",
"EENg": "x^6 - 16*x^5 + 97*x^4 - 278*x^3 + 376*x^2 - 192*x",
"EC\\w": "x^6 - 16*x^5 + 95*x^4 - 256*x^3 + 304*x^2 - 120*x",
"E@^W": "x^6 - 16*x^5 + 95*x^4 - 256*x^3 + 305*x^2 - 126*x",
"E_]w": "x^6 - 16*x^5 + 96*x^4 - 268*x^3 + 348*x^2 - 168*x",
"E@vw": "x^6 - 18*x^5 + 122*x^4 - 386*x^3 + 568*x^2 - 312*x",
"E@^o": "x^6 - 16*x^5 + 96*x^4 - 266*x^3 + 332*x^2 - 144*x",
"E@]w": "x^6 - 16*x^5 + 94*x^4 - 244*x^3 + 264*x^2 - 96*x",
"E@^w": "x^6 - 18*x^5 + 121*x^4 - 372*x^3 + 508*x^2 - 240*x",
"E@~o": "x^6 - 18*x^5 + 124*x^4 - 408*x^3 + 640*x^2 - 384*x",
"E@~w": "x^6 - 20*x^5 + 152*x^4 - 544*x^3 + 912*x^2 - 576*x",
"EBYW": {
"number": "x^6 - 14*x^5 + 73*x^4 - 176*x^3 + 192*x^2 - 72*x",
"comment": "It has the same Laplacian polynomial as EK`w."
},
"EBYw": "x^6 - 16*x^5 + 96*x^4 - 266*x^3 + 333*x^2 - 144*x",
"EBZw": "x^6 - 18*x^5 + 122*x^4 - 384*x^3 + 549*x^2 - 270*x",
"EIe_": {
"number": "x^6 - 12*x^5 + 54*x^4 - 112*x^3 + 105*x^2 - 36*x",
"comment": "This is the cycle $C_6$."
},
"EKNG": "x^6 - 14*x^5 + 74*x^4 - 184*x^3 + 213*x^2 - 90*x",
"EHUW": "x^6 - 14*x^5 + 73*x^4 - 174*x^3 + 185*x^2 - 66*x",
"EoLW": "x^6 - 14*x^5 + 74*x^4 - 182*x^3 + 205*x^2 - 84*x",
"E@]o": "x^6 - 14*x^5 + 72*x^4 - 164*x^3 + 156*x^2 - 48*x",
"EA]o": "x^6 - 14*x^5 + 73*x^4 - 174*x^3 + 184*x^2 - 66*x",
"EaMw": "x^6 - 16*x^5 + 97*x^4 - 276*x^3 + 364*x^2 - 174*x",
"EBhw": "x^6 - 16*x^5 + 96*x^4 - 264*x^3 + 320*x^2 - 126*x",
"EBjW": "x^6 - 16*x^5 + 97*x^4 - 276*x^3 + 365*x^2 - 180*x",
"EBjw": "x^6 - 18*x^5 + 123*x^4 - 396*x^3 + 595*x^2 - 330*x",
"EImo": "x^6 - 16*x^5 + 98*x^4 - 288*x^3 + 405*x^2 - 216*x",
"EJYW": {
"number": "x^6 - 16*x^5 + 97*x^4 - 274*x^3 + 348*x^2 - 144*x",
"comment": "It has the same Laplacian polynomial as E`NW."
},
"EIno": "x^6 - 18*x^5 + 125*x^4 - 418*x^3 + 672*x^2 - 414*x",
"EImw": "x^6 - 18*x^5 + 124*x^4 - 406*x^3 + 627*x^2 - 360*x",
"EInw": "x^6 - 20*x^5 + 154*x^4 - 568*x^3 + 999*x^2 - 666*x",
"EI]w": "x^6 - 18*x^5 + 123*x^4 - 392*x^3 + 564*x^2 - 270*x",
"EB]w": "x^6 - 18*x^5 + 122*x^4 - 380*x^3 + 523*x^2 - 240*x",
"EB^w": "x^6 - 20*x^5 + 152*x^4 - 538*x^3 + 855*x^2 - 450*x",
"EHuw": "x^6 - 18*x^5 + 124*x^4 - 406*x^3 + 628*x^2 - 366*x",
"EBnW": "x^6 - 18*x^5 + 123*x^4 - 394*x^3 + 585*x^2 - 324*x",
"EBnw": "x^6 - 20*x^5 + 153*x^4 - 554*x^3 + 939*x^2 - 594*x",
"EB~w": "x^6 - 22*x^5 + 187*x^4 - 762*x^3 + 1476*x^2 - 1080*x",
"EFz_": {
"number": "x^6 - 18*x^5 + 126*x^4 - 432*x^3 + 729*x^2 - 486*x",
"comment": "This is the complete bipartite graph $K_{3,3}$."
},
"Es\\w": "x^6 - 20*x^5 + 156*x^4 - 594*x^3 + 1107*x^2 - 810*x",
"EB~o": "x^6 - 20*x^5 + 155*x^4 - 580*x^3 + 1044*x^2 - 720*x",
"EFzw": "x^6 - 22*x^5 + 189*x^4 - 792*x^3 + 1620*x^2 - 1296*x",
"EF~w": "x^6 - 24*x^5 + 225*x^4 - 1026*x^3 + 2268*x^2 - 1944*x",
"EPTW": "x^6 - 14*x^5 + 73*x^4 - 172*x^3 + 172*x^2 - 48*x",
"E`NG": "x^6 - 14*x^5 + 74*x^4 - 180*x^3 + 189*x^2 - 54*x",
"E`NW": {
"number": "x^6 - 16*x^5 + 97*x^4 - 274*x^3 + 348*x^2 - 144*x",
"comment": "It has the same Laplacian polynomial as EJYW."
},
"E`Lw": "x^6 - 16*x^5 + 96*x^4 - 262*x^3 + 304*x^2 - 96*x",
"E`Nw": "x^6 - 18*x^5 + 123*x^4 - 394*x^3 + 576*x^2 - 288*x",
"EJeg": "x^6 - 16*x^5 + 98*x^4 - 284*x^3 + 381*x^2 - 180*x",
"EJqw": "x^6 - 18*x^5 + 125*x^4 - 416*x^3 + 656*x^2 - 384*x",
"EK\\w": "x^6 - 18*x^5 + 123*x^4 - 390*x^3 + 548*x^2 - 240*x",
"EK]w": "x^6 - 18*x^5 + 124*x^4 - 404*x^3 + 611*x^2 - 330*x",
"E`]w": "x^6 - 18*x^5 + 124*x^4 - 404*x^3 + 612*x^2 - 336*x",
"EK^w": "x^6 - 20*x^5 + 154*x^4 - 566*x^3 + 980*x^2 - 624*x",
"EK~o": "x^6 - 20*x^5 + 156*x^4 - 592*x^3 + 1088*x^2 - 768*x",
"EK~w": "x^6 - 22*x^5 + 188*x^4 - 776*x^3 + 1536*x^2 - 1152*x",
"EBn_": "x^6 - 16*x^5 + 98*x^4 - 286*x^3 + 397*x^2 - 210*x",
"E`]o": "x^6 - 16*x^5 + 98*x^4 - 284*x^3 + 384*x^2 - 192*x",
"ELpw": "x^6 - 18*x^5 + 125*x^4 - 416*x^3 + 659*x^2 - 396*x",
"ELrw": {
"number": "x^6 - 20*x^5 + 155*x^4 - 580*x^3 + 1045*x^2 - 726*x",
"comment": "This is the wheel $W_6$."
},
"ELv_": "x^6 - 18*x^5 + 126*x^4 - 428*x^3 + 705*x^2 - 450*x",
"Ek]w": "x^6 - 20*x^5 + 156*x^4 - 592*x^3 + 1091*x^2 - 780*x",
"EJnW": "x^6 - 20*x^5 + 155*x^4 - 578*x^3 + 1027*x^2 - 684*x",
"Ebnw": "x^6 - 22*x^5 + 189*x^4 - 790*x^3 + 1601*x^2 - 1254*x",
"EL~w": "x^6 - 24*x^5 + 226*x^4 - 1040*x^3 + 2328*x^2 - 2016*x",
"ER^W": "x^6 - 20*x^5 + 155*x^4 - 578*x^3 + 1028*x^2 - 690*x",
"EJmw": "x^6 - 20*x^5 + 154*x^4 - 564*x^3 + 965*x^2 - 600*x",
"EJnw": "x^6 - 22*x^5 + 188*x^4 - 774*x^3 + 1517*x^2 - 1110*x",
"EJ]w": "x^6 - 20*x^5 + 153*x^4 - 548*x^3 + 880*x^2 - 450*x",
"EJ^w": "x^6 - 22*x^5 + 186*x^4 - 740*x^3 + 1325*x^2 - 750*x",
"EJ~w": "x^6 - 24*x^5 + 225*x^4 - 1022*x^3 + 2220*x^2 - 1800*x",
"EN~w": "x^6 - 26*x^5 + 267*x^4 - 1350*x^3 + 3348*x^2 - 3240*x",
"EL~o": "x^6 - 22*x^5 + 190*x^4 - 804*x^3 + 1664*x^2 - 1344*x",
"EJ~o": "x^6 - 22*x^5 + 189*x^4 - 788*x^3 + 1580*x^2 - 1200*x",
"Ejmw": "x^6 - 22*x^5 + 190*x^4 - 804*x^3 + 1665*x^2 - 1350*x",
"ENzw": "x^6 - 24*x^5 + 227*x^4 - 1056*x^3 + 2412*x^2 - 2160*x",
"E]~o": "x^6 - 24*x^5 + 228*x^4 - 1072*x^3 + 2496*x^2 - 2304*x",
"E]~w": "x^6 - 26*x^5 + 268*x^4 - 1368*x^3 + 3456*x^2 - 3456*x",
"E^~w": "x^6 - 28*x^5 + 312*x^4 - 1728*x^3 + 4752*x^2 - 5184*x",
"E~~w": {
"number": "x^6 - 30*x^5 + 360*x^4 - 2160*x^3 + 6480*x^2 - 7776*x",
"comment": "This is the complete graph $K_6$."
}
}
},
"T536": {
"Title": "Distance spectral radii of connected graphs",
"Numbers": {
"A_": {
"number": "1",
"comment": "This is the complete graph $K_2$."
},
"BW": {
"number": "2.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576",
"comment": "This is the path $P_3$."
},
"Bw": {
"number": "2",
"comment": "This is the complete graph $K_3$."
},
"CF": {
"number": "4.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611",
"comment": "This is the star $K_{1,3}$."
},
"CL": {
"number": "5.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295",
"comment": "This is the path $P_4$."
},
"CN": "4.099647729675863158675706515466927602319390690346127273803750827689158883277832141800557412376273377",
"C]": {
"number": "4",
"comment": "This is the cycle $C_4$."
},
"C^": {
"number": "3.561552812808830274910704927987038512573599612686810217199316786547477173168810796793931825405342148",
"comment": "This is the diamond graph."
},
"C~": {
"number": "3",
"comment": "This is the complete graph $K_4$."
},
"D?{": {
"number": "6.605551275463989293119221267470495946251296573845246212710453056227166948293010445204619082018490718",
"comment": "This is the star $K_{1,4}$."
},
"D@s": "7.459294270913428162433205013554037487795044817926997559974706940831408777556070525950059733540536784",
"D@{": "6.176436314473940161400052549570315708794081228551998666981668935370174219918156562654580826816612427",
"DBg": {
"number": "8.288215751900178950371746248678562671344866133244798081722290871373455831531987299124399967819960068",
"comment": "This is the path $P_5$."
},
"DBk": {
"number": "6.637519478750677316156696280583774841913522961674326189403940151469115887764029425768788465127360627",
"comment": "This is the bull graph."
},
"DB{": {
"number": "5.759554956003028132093414842872558986575440473758742381839594459818255139180555632039823391844505605",
"comment": "This is the dart graph."
},
"DFw": {
"number": "5.645751311064590590501615753639260425710259183082450180368334459201068823230283627760392886474543611",
"comment": "This is the complete bipartite graph $K_{2,3}$."
},
"DF{": "5.372281323269014329925305734109464659110132228991396183849938735282950360728702313513562682798394061",
"DIk": "6.538120588189747960021694933395472939464804936318637666068341087169715470546716525799027254611097423",
"DJk": "6.216093478037556083123107398743380015589001798356771276356918801906444630229964359527550111414434658",
"DJ{": "5.344101953856734878344645526995957078769595652192982726544581295970003080172280804128246391599700512",
"DK[": "7.008607590101196064398798693505261146041916386456766158552076972131312213525592252405542238541492250",
"DK{": {
"number": "5.701562118716424343244108837310906632260210066310509442764636313334091379098438037144677151124934982",
"comment": "This is the butterfly graph."
},
"DLo": {
"number": "6",
"comment": "This is the cycle $C_5$."
},
"DL{": "5.292579941397382988462873482806539078290757213908552043130256646814500014700555254282946181667201384",
"DNw": "5.223861091336562290712446137350899199853715513366496509967119589905337488351075154918039518654610235",
"DN{": {
"number": "4.901799616122767589724438396507893410073196227938872470547867866464590020260870190406101946069017205",
"comment": "This is the house X graph."
},
"D]{": {
"number": "4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145",
"comment": "This is the wheel $W_5$."
},
"D^{": "4.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235",
"Dbk": {
"number": "5.635128415952847683392628140931560746756809197267229533769751009289379978389996138031317081911266247",
"comment": "This is the house graph."
},
"D~{": {
"number": "4",
"comment": "This is the complete graph $K_5$."
},
"E?Bw": {
"number": "8.582575694955840006588047193728008488984456576767971902607242123906868425547770886604361559493445033",
"comment": "This is the star $K_{1,5}$."
},
"E?Fg": "9.670211018734627189228617714528282628234663854278607846356762400771157566469275270689222319567798188",
"E?Fw": "8.229715652120553373467794397941149509198491240148617129252458453796052353354843446254494038403950360",
"E?NG": "10",
"E?NW": "8.990900305708534201682537819141873414188355327914355847197855129765655045969610028581973813062396994",
"E?Nw": "7.888793405231792297363150391889215758177958270756000349894833698927500627537168790783908560826317915",
"E?]o": "9.656854249492380195206754896838792314278687501507792292706718951962929913848428155401550137310566291",
"E?]w": "8.288215751900178950371746248678562671344866133244798081722290871373455831531987299124399967819960068",
"E?^o": "8.146800595958766194490816839838282450465894536561965486208464511578214144547629288282298072338737691",
"E?^w": "7.567276711409522560638865940600100219492452075976280276012812367876058127367819645952942247081347434",
"E?~o": {
"number": "7.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151",
"comment": "This is the complete bipartite graph $K_{2,4}$."
},
"E?~w": "7.274917217635374848618342403473058529111097352311690069149313452855360976958905597791226037073761986",
"E@FG": "11.05878486949216080547708779351584379966310305845069350967229464688083704469526902060289649616480594",
"E@JW": "8.898979485566356196394568149411782783931894961313340256865385134501920754914630053079718866209280470",
"E@NW": "8.637759361129662102148483197888900557899877717903538138926609387989050091045836853379650620030344859",
"E@Nw": "7.554564277386919362774449880271303396069502167737382725675635821475182316982946830625816662751923839",
"E@QW": "10.74240472910472916908291044199956316489767941661348617768463075654630879070548453925384739483275944",
"E@Rw": "7.852644393530912526622150298105355938395854109001691956865370786644459093497515401988188096639631198",
"E@UW": "9.385164807134504031250710491540329556295120161644788837680388670016645962827658692876633781679835484",
"E@Vw": "7.522196964570767434938017382530367343908765557314464055986180481639900169074654448493116624818980613",
"E@YO": {
"number": "12.10931150391298370620983766694289660538155474524907660481956459561042424295026087351713470158328136",
"comment": "This is the path $P_6$."
},
"E@YW": "9.260584690415349079941152017645486242816527934321247053809457608026135454885875756191557845981223551",
"E@]o": "9.424428900898052360873105707458812240848068143721382085861577264918220291853839315046600392057712881",
"E@]w": "7.977672660429449279764716873787928499326969873639479215975105361487284104826134408415222711311780577",
"E@^W": "7.915087677135012970061129449056767430777656847485019486145990871358703197793844751966320615638428774",
"E@^o": "7.837008155390812238535733671293445267586836945407681360359152134571838306218350888119689783595255319",
"E@^w": "7.205654436332271207049080263375133036919551090641558966218863569096848957606463334664232094934194396",
"E@`w": "9.315409046419373184893656775502349769848285107512745154067090098118720941048426274100938186707398030",
"E@ow": "9.608834688648652451303067349132198607973923492343224453229248580100739725549206494105380553515431137",
"E@vw": "7.174195006007347579837472047308365407724499255754429636007715560469229558775606301306215631644203245",
"E@~o": "7.095932280272493380194238705409093674501781282808419785925260536923723757542568790887773730370803540",
"E@~w": "6.885832006687951845298617196510203419300208176888623062404017188422368780139538943149176038101651526",
"EAMw": "8.663246504856263300666169628885890720463144153810259657011094709904045947534893129106636025922194529",
"EA]o": "8.523734714584910083312132158013749129335582764326241938854043089645221103175034341964588604400868337",
"EBYW": "8.570406330223831502468254715068369576724559332781366595508513465679279688774753347040297470339196649",
"EBYw": "7.875941589600097236287382761030896550438626384597017222910468321248203970957311912559430830689098689",
"EBZw": "7.164771202832824124194327792461065992272902281879330224052727465413670664857358440064146620731333210",
"EB]w": "7.623457321061652831168424141965573978090931624940271961176694868303324707497069048982412131742461810",
"EB^w": "6.860059361327185542723613553731716743754134896076227544732572170000666315489887295959285164600450690",
"EBhw": "8.272250914089625927804783360695648020260392052565663416024697921683990861689299591897777375885917437",
"EBjW": "7.400194923348946833685658168794736823895494067313727301308503512736347822741996805631708297816311873",
"EBjw": "7.128692311208703592146714993807059432664591884595827086690476311597575623378057195364750153780587838",
"EBnW": "7.123105625617660549821409855974077025147199225373620434398633573094954346337621593587863650810684297",
"EBn_": "7.359439711938180839365460011361866893778914411980412204300563768562098985400345353122808968343431066",
"EBnw": "6.828864774980929667905794366396536913476902363142080109320691794241148731583231915837499324378793120",
"EB~o": "6.743961887574750560850726629333186485842192601379283958148378190662678590218497610697151982677103159",
"EB~w": "6.510884131908910698178790570255454006911758511258012788183871982313161967686961711815014791604588480",
"ECXw": "8.969444711075949187591803286832929014459689648044004886496686843344524801670762746775813160170800768",
"EC\\w": "8.356914314274284520633788500018635632922613917502585330405527486544464846493156466246985726096092680",
"EENg": "7.405798995206611286163028458593608255754796110218969573393049779489386977560897877486859745398078716",
"EFz_": {
"number": "7",
"comment": "This is the complete bipartite graph $K_{3,3}$."
},
"EFzw": "6.418832675970042219334093825312152084961129431382511695838707815789712255468079795063241121849676217",
"EF~w": "6.162277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295",
"EGNW": "8.182150654236994475006362564002466165452199282419406893340735957376894832171118493402620666659105014",
"EGcw": "10.05477840878375749098355587496485442082587932527640572682842509515272871544867374847162888288268477",
"EGdo": "10.66796298636623923883474149716228219201580045250911957964717992376108726071992101996794722852038977",
"EGdw": "8.593600650603732712488925120145674114913733798787511686783007464226811360797879460116751541191782581",
"EHQW": "8.821870029750567683240300058048975807445655340022576146673553738590704358882348733338351930605328784",
"EHUW": "8.554312326447793500058619371691290452220079633268971054563990788635106731198900620196099842691574506",
"EHuw": "7.083495513981662886264185120581187826780476885072403495674916912934891160035103909284072877382351144",
"EI]w": "7.545472388670554997381672416771862308001230594101176173834728808312117060172207142171041082468624519",
"EI_w": "9.971258158604132034753567894830528555021754610489722977025589796264347896530968473823813833624636135",
"EIe_": {
"number": "9",
"comment": "This is the cycle $C_6$."
},
"EImo": "7.772001872658765583935824163119853217297227664766641120954325626885824409663649190540486015053504715",
"EImw": "7.516940184447188676915949294280308012351510067955662026145938301729999121467464606543094727642148027",
"EIno": "7.042503530717760831210248952032156402742408249401329556630938164777930363869253658043537069779915059",
"EInw": "6.788444596718009300536725196800827188355907389777955378449343898800245184514965516970038326243553744",
"EJYW": "8.203702666941517294362591787347178886893961463200912762026189940200631176450925847270379980556515645",
"EJ]w": "7.267499761734287857670893062963125654616131142707945966861439253323515806256439950589968973651262171",
"EJ^w": "6.524199596856449985769506149946415873052548914463208417599766061899525407490272084360285122282475577",
"EJeg": "7.772001872658765583935824163119853217297227664766641120954325626885824409663649190540486015053504715",
"EJmw": "7.242640687119285146405066172629094235709015626130844219530039213972197435386321116551162602982924718",
"EJnW": "6.736331326907798122051111778672633446835383700962481000533301932297189703849017417117537207264034748",
"EJnw": "6.452337379650123670663474484614945418139359611564221793583074680875098528145714518152341167915423020",
"EJqw": "7.041841636255810501233685160551103535020633327871181677084970773276739137817151795314216980947121903",
"EJ~o": "6.401723327102481564331370877475360917312881192494227122371657361676311148109009223467764063761012468",
"EJ~w": "6.142457360426860982974371175813316052427099711323624839031602935476764350819921712950679474132003467",
"EKNG": "8.424428900898052360873105707458812240848068143721382085861577264918220291853839315046600392057712881",
"EK\\w": "7.967588056209378452379670823870465822647134502183696635589026556713721676837242005206601091069973303",
"EK]w": "7.516538485191787521369500838674010010643324846946545787137978201021031986419442426823993247805965004",
"EK^w": "6.783023515112379064477740252122012723352229914482963301860425766504531615296784490776230865593238407",
"EK`w": "8.496968025764426841840223787923295483248967624928648796254298262586636405124665244290543730447327654",
"EK~o": "6.701562118716424343244108837310906632260210066310509442764636313334091379098438037144677151124934982",
"EK~w": "6.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151",
"ELpw": "7.037188190781964977296205363519877137417380080077899804807933069751966776283525037252604997699543249",
"ELrw": {
"number": "6.741657386773941385583748732316549301756019807778726946303745467320035156306939027976809895194379572",
"comment": "This is the wheel $W_6$."
},
"ELv_": "7",
"EL~o": "6.364681864288780538128581263924837610260550074327576979431947169317669756172975378598305759821536095",
"EL~w": "6.101160353032533717661193469040595983297214967752215127390605539111030738594108655345324258820149902",
"ENzw": "6.047904976996737742163716383969281044161339546139485124629625582172482823319022836931898636505573337",
"EN~w": "5.756591308025603270998975945787073160501526529555139398775839250829654611721784750796198757384691739",
"EPTW": "9.696375622397740570346133988988181739717002740718336754927416262550646170989250684637466527806217463",
"ER^W": "6.732148054190623977865415792967958060083604475806641809284701122616795658404697927235856415729005485",
"E]~o": "6",
"E]~w": "5.701562118716424343244108837310906632260210066310509442764636313334091379098438037144677151124934982",
"E^~w": "5.372281323269014329925305734109464659110132228991396183849938735282950360728702313513562682798394061",
"E_Nw": "7.486354991477194587810544092582420298616950775046712703538138233020529789232377306834588110371089787",
"E_]o": "7.695199486633531488042820650897823521892277875555972800468500029050274437616837088363272588210083867",
"E_]w": "7.441739268043944811016935379813314111757068138723987060507981385917704801447787608776867850965759442",
"E`Lw": "8.627894657894644955633313621753252388763130899973028371585766424770653155835189940179749238994859170",
"E`NG": "9.196152422706631880582339024517617100828415761431141884167420938355799050726400111243438560271745727",
"E`NW": "8.216990566028301905660330188811320358491811316707560660331490724490011454792559029270513493445192052",
"E`Nw": "7.124597730399937290206538842323205202645868279920251608002838682507189555301569230878031761381501038",
"E`]o": "7.355350340869579713899483992555466160301405405663530334667860235581023044096891224821228503859471970",
"E`]w": "7.074673582515124999909700639207229149655768890305131583284462717931675053894483650282604193044552183",
"EaMw": "7.804345859672770990287182619046831708363517408070254310177462709533055198198415686981136961576371635",
"Ebnw": "6.411381821997608307153884791711556675941324540186737023031897559555320696387555301220295702281961248",
"Ejmw": "6.358898943540673552236981983859615659137003925232444936890344138159557328203158085656159155851944527",
"Ek]w": "6.695265165690463668457504628088737706837499318954419150171925630980501052614879528546353923469763702",
"EoLW": "8.074065501057913616329703499494772306880808815383812708551950813010665461034138872864632674844301072",
"Es\\w": "6.696099695490513414298842649161240311132292160789810095310187181242567553034343850100182572849175248",
"E~~w": {
"number": "5",
"comment": "This is the complete graph $K_6$."
}
}
},
"T537": {
"Title": "Distance characteristic polynomials of connected graphs",
"Numbers": {
"A_": {
"number": "x^2 - 1",
"comment": "This is the complete graph $K_2$."
},
"BW": {
"number": "x^3 - 6*x - 4",
"comment": "This is the path $P_3$."
},
"Bw": {
"number": "x^3 - 3*x - 2",
"comment": "This is the complete graph $K_3$."
},
"CF": {
"number": "x^4 - 15*x^2 - 28*x - 12",
"comment": "This is the star $K_{1,3}$."
},
"CL": {
"number": "x^4 - 20*x^2 - 32*x - 12",
"comment": "This is the path $P_4$."
},
"CN": {
"number": "x^4 - 12*x^2 - 18*x - 7"
},
"C]": {
"number": "x^4 - 12*x^2 - 16*x",
"comment": "This is the cycle $C_4$."
},
"C^": {
"number": "x^4 - 9*x^2 - 12*x - 4",
"comment": "This is the diamond graph."
},
"C~": {
"number": "x^4 - 6*x^2 - 8*x - 3",
"comment": "This is the complete graph $K_4$."
},
"D?{": {
"number": "x^5 - 28*x^3 - 88*x^2 - 96*x - 32",
"comment": "This is the star $K_{1,4}$."
},
"D@s": {
"number": "x^5 - 38*x^3 - 116*x^2 - 112*x - 32"
},
"D@{": {
"number": "x^5 - 25*x^3 - 70*x^2 - 66*x - 20"
},
"DBg": {
"number": "x^5 - 50*x^3 - 140*x^2 - 120*x - 32",
"comment": "This is the path $P_5$."
},
"DBk": {
"number": "x^5 - 30*x^3 - 82*x^2 - 72*x - 20",
"comment": "This is the bull graph."
},
"DB{": {
"number": "x^5 - 22*x^3 - 56*x^2 - 46*x - 12",
"comment": "This is the dart graph."
},
"DFw": {
"number": "x^5 - 22*x^3 - 52*x^2 - 24*x + 16",
"comment": "This is the complete bipartite graph $K_{2,3}$."
},
"DF{": {
"number": "x^5 - 19*x^3 - 46*x^2 - 36*x - 8"
},
"DIk": {
"number": "x^5 - 30*x^3 - 76*x^2 - 48*x"
},
"DJk": {
"number": "x^5 - 27*x^3 - 64*x^2 - 50*x - 12"
},
"DJ{": {
"number": "x^5 - 19*x^3 - 44*x^2 - 36*x - 10"
},
"DK[": {
"number": "x^5 - 35*x^3 - 88*x^2 - 74*x - 20"
},
"DK{": {
"number": "x^5 - 22*x^3 - 52*x^2 - 43*x - 12",
"comment": "This is the butterfly graph."
},
"DLo": {
"number": "x^5 - 25*x^3 - 60*x^2 - 35*x - 6",
"comment": "This is the cycle $C_5$."
},
"DL{": {
"number": "x^5 - 19*x^3 - 42*x^2 - 29*x - 6"
},
"DNw": {
"number": "x^5 - 19*x^3 - 40*x^2 - 18*x + 4"
},
"DN{": {
"number": "x^5 - 16*x^3 - 34*x^2 - 25*x - 6",
"comment": "This is the house X graph."
},
"D]{": {
"number": "x^5 - 16*x^3 - 32*x^2 - 16*x",
"comment": "This is the wheel $W_5$."
},
"D^{": {
"number": "x^5 - 13*x^3 - 26*x^2 - 18*x - 4"
},
"Dbk": {
"number": "x^5 - 22*x^3 - 50*x^2 - 28*x",
"comment": "This is the house graph."
},
"D~{": {
"number": "x^5 - 10*x^3 - 20*x^2 - 15*x - 4",
"comment": "This is the complete graph $K_5$."
},
"E?Bw": {
"number": "x^6 - 45*x^4 - 200*x^3 - 360*x^2 - 288*x - 80",
"comment": "This is the star $K_{1,5}$."
},
"E?Fg": {
"number": "x^6 - 60*x^4 - 272*x^3 - 468*x^2 - 336*x - 80"
},
"E?Fw": {
"number": "x^6 - 42*x^4 - 174*x^3 - 285*x^2 - 204*x - 52"
},
"E?NG": {
"number": "x^6 - 65*x^4 - 296*x^3 - 504*x^2 - 352*x - 80"
},
"E?NW": {
"number": "x^6 - 52*x^4 - 218*x^3 - 345*x^2 - 228*x - 52"
},
"E?Nw": {
"number": "x^6 - 39*x^4 - 152*x^3 - 228*x^2 - 144*x - 32"
},
"E?]o": {
"number": "x^6 - 64*x^4 - 248*x^3 - 320*x^2 - 128*x"
},
"E?]w": {
"number": "x^6 - 44*x^4 - 172*x^3 - 252*x^2 - 152*x - 32"
},
"E?^o": {
"number": "x^6 - 44*x^4 - 160*x^3 - 180*x^2 - 16*x + 48"
},
"E?^w": {
"number": "x^6 - 36*x^4 - 134*x^3 - 189*x^2 - 108*x - 20"
},
"E?~o": {
"number": "x^6 - 36*x^4 - 128*x^3 - 144*x^2 + 64",
"comment": "This is the complete bipartite graph $K_{2,4}$."
},
"E?~w": {
"number": "x^6 - 33*x^4 - 120*x^3 - 168*x^2 - 96*x - 16"
},
"E@FG": {
"number": "x^6 - 84*x^4 - 368*x^3 - 580*x^2 - 368*x - 80"
},
"E@JW": {
"number": "x^6 - 52*x^4 - 208*x^3 - 288*x^2 - 128*x"
},
"E@NW": {
"number": "x^6 - 49*x^4 - 188*x^3 - 268*x^2 - 160*x - 32"
},
"E@Nw": {
"number": "x^6 - 36*x^4 - 132*x^3 - 189*x^2 - 120*x - 28"
},
"E@QW": {
"number": "x^6 - 77*x^4 - 356*x^3 - 572*x^2 - 368*x - 80"
},
"E@Rw": {
"number": "x^6 - 39*x^4 - 148*x^3 - 217*x^2 - 140*x - 33"
},
"E@UW": {
"number": "x^6 - 57*x^4 - 248*x^3 - 384*x^2 - 240*x - 52"
},
"E@Vw": {
"number": "x^6 - 36*x^4 - 130*x^3 - 174*x^2 - 94*x - 17"
},
"E@YO": {
"number": "x^6 - 105*x^4 - 448*x^3 - 648*x^2 - 384*x - 80",
"comment": "This is the path $P_6$."
},
"E@YW": {
"number": "x^6 - 57*x^4 - 232*x^3 - 304*x^2 - 128*x"
},
"E@]o": {
"number": "x^6 - 61*x^4 - 228*x^3 - 304*x^2 - 168*x - 32"
},
"E@]w": {
"number": "x^6 - 41*x^4 - 152*x^3 - 212*x^2 - 128*x - 28"
},
"E@^W": {
"number": "x^6 - 41*x^4 - 146*x^3 - 188*x^2 - 98*x - 17"
},
"E@^o": {
"number": "x^6 - 41*x^4 - 140*x^3 - 152*x^2 - 40*x + 12"
},
"E@^w": {
"number": "x^6 - 33*x^4 - 114*x^3 - 149*x^2 - 84*x - 17"
},
"E@`w": {
"number": "x^6 - 57*x^4 - 236*x^3 - 360*x^2 - 232*x - 52"
},
"E@ow": {
"number": "x^6 - 62*x^4 - 250*x^3 - 373*x^2 - 236*x - 52"
},
"E@vw": {
"number": "x^6 - 33*x^4 - 112*x^3 - 138*x^2 - 64*x - 8"
},
"E@~o": {
"number": "x^6 - 33*x^4 - 108*x^3 - 108*x^2 + 32"
},
"E@~w": {
"number": "x^6 - 30*x^4 - 100*x^3 - 127*x^2 - 68*x - 12"
},
"EAMw": {
"number": "x^6 - 49*x^4 - 192*x^3 - 273*x^2 - 160*x - 32"
},
"EA]o": {
"number": "x^6 - 49*x^4 - 176*x^3 - 209*x^2 - 80*x"
},
"EBYW": {
"number": "x^6 - 49*x^4 - 184*x^3 - 216*x^2 - 32*x + 48"
},
"EBYw": {
"number": "x^6 - 41*x^4 - 144*x^3 - 165*x^2 - 44*x + 12"
},
"EBZw": {
"number": "x^6 - 33*x^4 - 112*x^3 - 132*x^2 - 48*x"
},
"EB]w": {
"number": "x^6 - 38*x^4 - 130*x^3 - 166*x^2 - 90*x - 17"
},
"EB^w": {
"number": "x^6 - 30*x^4 - 98*x^3 - 121*x^2 - 64*x - 12"
},
"EBhw": {
"number": "x^6 - 46*x^4 - 160*x^3 - 199*x^2 - 100*x - 17"
},
"EBjW": {
"number": "x^6 - 36*x^4 - 120*x^3 - 133*x^2 - 48*x"
},
"EBjw": {
"number": "x^6 - 33*x^4 - 108*x^3 - 127*x^2 - 60*x - 9"
},
"EBnW": {
"number": "x^6 - 33*x^4 - 108*x^3 - 123*x^2 - 54*x - 8"
},
"EBn_": {
"number": "x^6 - 36*x^4 - 118*x^3 - 114*x^2 - 10*x + 7"
},
"EBnw": {
"number": "x^6 - 30*x^4 - 96*x^3 - 112*x^2 - 54*x - 9"
},
"EB~o": {
"number": "x^6 - 30*x^4 - 92*x^3 - 84*x^2 + 16"
},
"EB~w": {
"number": "x^6 - 27*x^4 - 84*x^3 - 98*x^2 - 48*x - 8"
},
"ECXw": {
"number": "x^6 - 54*x^4 - 204*x^3 - 280*x^2 - 160*x - 32"
},
"EC\\w": {
"number": "x^6 - 46*x^4 - 170*x^3 - 229*x^2 - 124*x - 20"
},
"EENg": {
"number": "x^6 - 36*x^4 - 122*x^3 - 129*x^2 - 12*x + 28"
},
"EFz_": {
"number": "x^6 - 33*x^4 - 104*x^3 - 72*x^2 + 96*x + 112",
"comment": "This is the complete bipartite graph $K_{3,3}$."
},
"EFzw": {
"number": "x^6 - 27*x^4 - 80*x^3 - 72*x^2 + 16"
},
"EF~w": {
"number": "x^6 - 24*x^4 - 72*x^3 - 81*x^2 - 36*x - 4"
},
"EGNW": {
"number": "x^6 - 44*x^4 - 162*x^3 - 201*x^2 - 80*x"
},
"EGcw": {
"number": "x^6 - 69*x^4 - 280*x^3 - 405*x^2 - 244*x - 52"
},
"EGdo": {
"number": "x^6 - 81*x^4 - 308*x^3 - 424*x^2 - 248*x - 52"
},
"EGdw": {
"number": "x^6 - 49*x^4 - 182*x^3 - 253*x^2 - 152*x - 33"
},
"EHQW": {
"number": "x^6 - 52*x^4 - 198*x^3 - 250*x^2 - 114*x - 17"
},
"EHUW": {
"number": "x^6 - 49*x^4 - 180*x^3 - 220*x^2 - 80*x"
},
"EHuw": {
"number": "x^6 - 33*x^4 - 106*x^3 - 108*x^2 - 22*x + 7"
},
"EI]w": {
"number": "x^6 - 38*x^4 - 124*x^3 - 136*x^2 - 48*x"
},
"EI_w": {
"number": "x^6 - 69*x^4 - 268*x^3 - 340*x^2 - 128*x"
},
"EIe_": {
"number": "x^6 - 57*x^4 - 200*x^3 - 144*x^2",
"comment": "This is the cycle $C_6$."
},
"EImo": {
"number": "x^6 - 41*x^4 - 136*x^3 - 120*x^2 + 32*x + 48"
},
"EImw": {
"number": "x^6 - 38*x^4 - 122*x^3 - 125*x^2 - 28*x + 12"
},
"EIno": {
"number": "x^6 - 33*x^4 - 104*x^3 - 93*x^2 + 12*x + 28"
},
"EInw": {
"number": "x^6 - 30*x^4 - 94*x^3 - 99*x^2 - 28*x + 4"
},
"EJYW": {
"number": "x^6 - 46*x^4 - 154*x^3 - 165*x^2 - 44*x + 12"
},
"EJ]w": {
"number": "x^6 - 35*x^4 - 110*x^3 - 132*x^2 - 68*x - 12"
},
"EJ^w": {
"number": "x^6 - 27*x^4 - 84*x^3 - 105*x^2 - 60*x - 13"
},
"EJeg": {
"number": "x^6 - 41*x^4 - 132*x^3 - 140*x^2 - 48*x"
},
"EJmw": {
"number": "x^6 - 35*x^4 - 108*x^3 - 125*x^2 - 60*x - 9"
},
"EJnW": {
"number": "x^6 - 30*x^4 - 90*x^3 - 88*x^2 - 24*x"
},
"EJnw": {
"number": "x^6 - 27*x^4 - 80*x^3 - 87*x^2 - 38*x - 5"
},
"EJqw": {
"number": "x^6 - 33*x^4 - 102*x^3 - 101*x^2 - 24*x + 7"
},
"EJ~o": {
"number": "x^6 - 27*x^4 - 78*x^3 - 72*x^2 - 12*x + 8"
},
"EJ~w": {
"number": "x^6 - 24*x^4 - 70*x^3 - 81*x^2 - 42*x - 8"
},
"EKNG": {
"number": "x^6 - 49*x^4 - 168*x^3 - 144*x^2"
},
"EK\\w": {
"number": "x^6 - 43*x^4 - 140*x^3 - 173*x^2 - 92*x - 17"
},
"EK]w": {
"number": "x^6 - 38*x^4 - 120*x^3 - 135*x^2 - 60*x - 9"
},
"EK^w": {
"number": "x^6 - 30*x^4 - 92*x^3 - 105*x^2 - 50*x - 8"
},
"EK`w": {
"number": "x^6 - 49*x^4 - 172*x^3 - 204*x^2 - 80*x"
},
"EK~o": {
"number": "x^6 - 30*x^4 - 88*x^3 - 79*x^2 - 8*x + 12"
},
"EK~w": {
"number": "x^6 - 27*x^4 - 80*x^3 - 93*x^2 - 48*x - 9"
},
"ELpw": {
"number": "x^6 - 33*x^4 - 102*x^3 - 97*x^2 - 24*x"
},
"ELrw": {
"number": "x^6 - 30*x^4 - 90*x^3 - 90*x^2 - 36*x - 5",
"comment": "This is the wheel $W_6$."
},
"ELv_": {
"number": "x^6 - 33*x^4 - 100*x^3 - 84*x^2"
},
"EL~o": {
"number": "x^6 - 27*x^4 - 76*x^3 - 63*x^2 - 4*x + 4"
},
"EL~w": {
"number": "x^6 - 24*x^4 - 68*x^3 - 72*x^2 - 32*x - 5"
},
"ENzw": {
"number": "x^6 - 24*x^4 - 66*x^3 - 59*x^2 - 12*x + 4"
},
"EN~w": {
"number": "x^6 - 21*x^4 - 58*x^3 - 63*x^2 - 30*x - 5"
},
"EPTW": {
"number": "x^6 - 66*x^4 - 238*x^3 - 309*x^2 - 168*x - 32"
},
"ER^W": {
"number": "x^6 - 30*x^4 - 90*x^3 - 86*x^2 - 18*x + 7"
},
"E]~o": {
"number": "x^6 - 24*x^4 - 64*x^3 - 48*x^2"
},
"E]~w": {
"number": "x^6 - 21*x^4 - 56*x^3 - 52*x^2 - 16*x"
},
"E^~w": {
"number": "x^6 - 18*x^4 - 48*x^3 - 51*x^2 - 24*x - 4"
},
"E_Nw": {
"number": "x^6 - 36*x^4 - 126*x^3 - 167*x^2 - 96*x - 20"
},
"E_]o": {
"number": "x^6 - 39*x^4 - 136*x^3 - 147*x^2 - 28*x + 4"
},
"E_]w": {
"number": "x^6 - 36*x^4 - 124*x^3 - 144*x^2 - 48*x"
},
"E`Lw": {
"number": "x^6 - 51*x^4 - 174*x^3 - 228*x^2 - 132*x - 28"
},
"E`NG": {
"number": "x^6 - 59*x^4 - 204*x^3 - 269*x^2 - 156*x - 33"
},
"E`NW": {
"number": "x^6 - 46*x^4 - 152*x^3 - 191*x^2 - 104*x - 20"
},
"E`Nw": {
"number": "x^6 - 33*x^4 - 106*x^3 - 135*x^2 - 78*x - 17"
},
"E`]o": {
"number": "x^6 - 36*x^4 - 116*x^3 - 120*x^2 - 44*x - 5"
},
"E`]w": {
"number": "x^6 - 33*x^4 - 104*x^3 - 112*x^2 - 40*x"
},
"EaMw": {
"number": "x^6 - 41*x^4 - 136*x^3 - 145*x^2 - 48*x"
},
"Ebnw": {
"number": "x^6 - 27*x^4 - 78*x^3 - 76*x^2 - 24*x"
},
"Ejmw": {
"number": "x^6 - 27*x^4 - 76*x^3 - 61*x^2 + 4*x + 15"
},
"Ek]w": {
"number": "x^6 - 30*x^4 - 88*x^3 - 75*x^2 - 4*x + 7"
},
"EoLW": {
"number": "x^6 - 44*x^4 - 150*x^3 - 162*x^2 - 66*x - 9"
},
"Es\\w": {
"number": "x^6 - 30*x^4 - 90*x^3 - 69*x^2 + 36*x + 44"
},
"E~~w": {
"number": "x^6 - 15*x^4 - 40*x^3 - 45*x^2 - 24*x - 5",
"comment": "This is the complete graph $K_6$."
}
}
}
}