Kirchhoff indices of connected graphs

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Numbers

$G$

$\mathrm{Kf}(G)$

comment: This is the complete graph $K_1$.

A_:

comment: This is the complete graph $K_2$.

BW:

comment: This is the path $P_3$.

Bw:

comment: This is the complete graph $K_3$.

CF:

comment: This is the star $K_{1,3}$.

CL:

comment: This is the path $P_4$.

CN:

19/3

C]:

comment: This is the cycle $C_4$.

C^:

comment: This is the diamond graph.

C~:

comment: This is the complete graph $K_4$.

D?{:

comment: This is the star $K_{1,4}$.

D@s:

D@{:

38/3

DIk:

23/2

DBk:

40/3

comment: This is the bull graph.

DB{:

39/4

comment: This is the dart graph.

DFw:

23/3

comment: This is the complete bipartite graph $K_{2,3}$.

DF{:

DBg:

comment: This is the path $P_5$.

DK[:

44/3

DK{:

28/3

comment: This is the butterfly graph.

DLo:

comment: This is the cycle $C_5$.

Dbk:

90/11

comment: This is the house graph.

DL{:

146/21

DJk:

41/4

DJ{:

17/2

DN{:

23/4

comment: This is the house X graph.

DNw:

77/12

D]{:

16/3

comment: This is the wheel $W_5$.

D^{:

14/3

D~{:

comment: This is the complete graph $K_5$.

E?Bw:

comment: This is the star $K_{1,5}$.

E?Fg:

E?Fw:

E@FG:

E?NG:

E@JW:

E?NW:

67/3

E?Nw:

35/2

E?]o:

E?^o:

46/3

E?]w:

E?^w:

71/5

E?~o:

23/2

comment: This is the complete bipartite graph $K_{2,4}$.

E?~w:

E@QW:

E@`w:

E@Rw:

E@YW:

83/4

EHQW:

E@UW:

EGNW:

EAMw:

149/8

E@Vw:

99/7

E@YO:

comment: This is the path $P_6$.

EI_w:

EGdo:

EGcw:

76/3

E@ow:

EK`w:

ECXw:

41/2

EGdw:

55/3

E_Nw:

27/2

E@NW:

37/2

E@Nw:

E_]o:

27/2

EENg:

47/4

EC\w:

76/5

E@^W:

305/21

E_]w:

87/7

E@vw:

142/13

E@^o:

83/6

E@]w:

33/2

E@^w:

127/10

E@~o:

E@~w:

19/2

EBYW:

EBYw:

111/8

EBZw:

61/5

EIe_:

35/2

comment: This is the cycle $C_6$.

EKNG:

71/5

EHUW:

185/11

EoLW:

205/14

E@]o:

39/2

EA]o:

184/11

EaMw:

364/29

EBhw:

320/21

EBjW:

73/6

EBjw:

119/11

EImo:

45/4

EJYW:

29/2

EIno:

224/23

EImw:

209/20

EInw:

EI]w:

188/15

EB]w:

523/40

EB^w:

57/5

EHuw:

628/61

EBnW:

65/6

EBnw:

313/33

EB~w:

41/5

EFz_:

comment: This is the complete bipartite graph $K_{3,3}$.

Es\w:

41/5

EB~o:

87/10

EFzw:

15/2

EF~w:

EPTW:

43/2

E`NG:

E`NW:

29/2

E`Lw:

E`Nw:

EJeg:

127/10

EJqw:

41/4

EK\w:

137/10

EK]w:

611/55

E`]w:

153/14

EK^w:

245/26

EK~o:

17/2

EK~w:

EBn_:

397/35

E`]o:

ELpw:

659/66

ELrw:

95/11

comment: This is the wheel $W_6$.

ELv_:

47/5

Ek]w:

1091/130

EJnW:

1027/114

Ebnw:

1601/209

EL~w:

97/14

ER^W:

1028/115

EJmw:

193/20

EJnw:

41/5

EJ]w:

176/15

EJ^w:

53/5

EJ~w:

37/5

EN~w:

31/5

EL~o:

52/7

EJ~o:

79/10

Ejmw:

37/5

ENzw:

67/10

E]~o:

13/2

E]~w:

E^~w:

11/2

E~~w:

comment: This is the complete graph $K_6$.

F??Fw:

comment: This is the star $K_{1,6}$.

F??Ng:

F??Nw:

94/3

F?CNG:

F??^G:

F??^o:

61/2

F??^W:

100/3

F??^w:

109/4

F??}O:

F??}o:

65/2

F??}W:

F??~o:

F??}w:

113/4

F??~w:

117/5

F?@|o:

77/3

F?@~o:

41/2

F?@|w:

119/5

F?@~w:

59/3

F?B~o:

82/5

comment: This is the complete bipartite graph $K_{2,5}$.

F?B~w:

F?CeW:

F@?Mw:

106/3

F?Cfw:

80/3

F@?]O:

F?Cmg:

F@?]W:

F@CeW:

F?CmW:

104/3

F@?^W:

284/11

F?G]w:

F?Cnw:

70/3

F@OKg:

F?DcW:

F@IAw:

69/2

F@AJo:

124/3

F@@Kw:

112/3

F?Dcw:

112/3

F@PDw:

155/6

F?_rw:

125/4

F@@Lw:

86/3

FG?^w:

271/12

F?C^G:

112/3

F?C^W:

115/4

F?C^w:

51/2

F@@ko:

156/5

F?O|o:

79/3

F?W\g:

302/11

FG?}o:

187/8

F?H[w:

59/2

F?Dkw:

59/2

F@@kw:

290/11

F?W^g:

333/16

F?O|w:

F?Dlw:

508/21

FG?}w:

303/14

F?Dnw:

509/26

F?O|_:

73/2

F@?}O:

130/3

F?`ro:

FGAZo:

167/6

F?O|g:

129/4

F@?}W:

88/3

F_?~o:

61/3

F?`rw:

137/5

FGAZw:

283/12

F_?~w:

281/15

F?C~O:

300/11

F@C]W:

123/4

F?C~o:

93/4

F?C~W:

506/21

F?C}w:

53/2

F?C~w:

433/20

F?`zo:

87/4

F?D~O:

341/16

F_@|o:

203/11

F?Fno:

347/21

F?`zw:

127/6

F?D~W:

517/26

F_@|w:

301/17

F?Fnw:

493/31

F?D|o:

287/12

F?D~o:

75/4

F?D|w:

441/20

F?D~w:

215/12

F?F~o:

293/20

F?F~w:

57/4

F?KuW:

79/3

F@CmW:

302/11

F?Kuw:

70/3

F?Kvw:

316/15

FICcW:

88/3

F@HSW:

154/5

F?_yo:

F@O^G:

362/15

FGG[w:

67/2

F@Okw:

300/11

FI?\W:

174/7

F@O[w:

61/2

F?StW:

300/11

FGG]w:

632/29

F?MRw:

526/21

F?LVW:

64/3

F?LVw:

214/11

FIAHo:

81/2

F?StG:

130/3

F@G]G:

73/2

F?oow:

118/3

FIa@w:

F?org:

149/4

FIAHw:

167/6

F?opw:

133/4

F?YPw:

92/3

Fo?Zw:

87/4

F?NBw:

295/12

F?NFw:

37/2

F@H[o:

82/3

F@TTW:

62/3

F@IZo:

149/6

F?Tto:

146/7

F@H[w:

143/6

F@H^o:

1274/69

F@H]w:

58/3

F@H\w:

328/15

F@Dmw:

1168/61

F@H^w:

52/3

Fo?yo:

F?L^?:

F?[uG:

195/8

F_LLg:

419/22

F?Uro:

678/29

F?]Rg:

357/16

F?svG:

206/11

FADlW:

650/29

FCDjW:

147/5

Fo?yw:

184/9

F?NPw:

536/21

FEG^W:

709/42

FCDnW:

647/36

F@FJw:

1116/55

F?drw:

553/26

F?NVW:

328/19

F@FNw:

1069/68

F@KuW:

295/12

F@G}w:

326/15

F?K}w:

112/5

F?K~w:

302/15

F?L^_:

104/5

FAEhw:

180/7

F?L\g:

143/6

F?L^G:

328/15

F?L[w:

F@D^W:

1166/61

F?S|w:

45/2

F?L^W:

178/9

F?L^w:

1772/99

F?NN_:

F?drW:

323/14

F?LuW:

341/16

F?K}W:

127/5

F?Lkw:

74/3

F?NNg:

169/10

F?NJw:

529/26

F?NNw:

127/8

F@J]o:

193/12

F@J^o:

427/30

F@J]w:

31/2

F@J^w:

1805/132

F@H}o:

1304/69

F?L~o:

517/30

F@H}w:

1954/111

F?L|w:

308/15

F?L~w:

82/5

F@Fmo:

255/16

F?s~g:

1187/80

F@Fmw:

1125/74

F?L}w:

1802/99

F?dzw:

56/3

F?N^W:

520/33

F?N^w:

1679/118

F?N~o:

197/15

F?N~w:

191/15

FBXcw:

F?\sw:

77/4

FIO|w:

50/3

F?\tw:

1037/60

F?\vw:

187/12

F?]v_:

328/21

F?lrg:

311/15

F?\tg:

1310/69

FBEmW:

148/9

F?L|o:

112/5

F?T|o:

1196/61

F?v`w:

50/3

F?lvg:

278/19

F?]rw:

2008/111

F?]uw:

166/11

F?]vw:

532/39

FIQ|o:

161/12

F?\~_:

256/15

F?]~_:

886/63

F?^vo:

162/13

FIQ|w:

77/6

F?^vw:

680/57

F?\~g:

95/6

F?\|w:

167/10

F?\~w:

F?^tw:

40/3

F?]}w:

156/11

F?]~w:

5462/429

F?^~w:

34/3

F?~v_:

23/2

comment: This is the complete bipartite graph $K_{3,4}$.

F?~vg:

131/12

F?^~o:

176/15

F?~vw:

52/5

F?~~w:

F@Q?w:

F@Q@w:

118/3

F@QBw:

92/3

F@QFw:

F@HSO:

comment: This is the path $P_7$.

F@O\G:

FA_pW:

FHQ?w:

F@OsW:

124/3

F@R@o:

142/3

F@QGw:

116/3

FG_qw:

328/11

FGEJg:

82/3

FG_Zg:

100/3

F`Q@w:

76/3

FA_hw:

F@QHw:

F`?^W:

698/33

FCHJw:

82/3

FGEJw:

73/3

F_Cnw:

56/3

FGC\W:

131/4

FGC^G:

98/3

FGC^W:

289/12

FGC\w:

59/2

FGC^w:

125/6

FK?}O:

64/3

FAMRW:

177/8

F@QZo:

68/3

F_W\g:

818/43

FA_xw:

132/5

FAIXw:

178/7

FCHiw:

590/21

FG_yw:

FGC}W:

644/29

FK?}W:

766/39

FA_~o:

284/17

F@`Zw:

275/13

F@QZw:

222/11

F_Dlw:

1296/73

F@Q^w:

142/9

F@LSW:

F@Maw:

295/12

FAStW:

338/15

FGD\o:

173/8

F@Lew:

305/16

F@_zw:

93/4

FAG}w:

1106/55

FGC}w:

557/28

F@O~w:

927/52

F`?}O:

116/3

FODZo:

109/4

F_K^G:

746/33

FGEZo:

313/12

FGEXw:

61/2

FGDkw:

149/6

F`?}W:

74/3

F`C^W:

223/12

FODZw:

513/20

F_C~W:

136/7

FGEZw:

131/6

F_C~w:

1019/60

F@`zo:

F@Q}o:

1452/85

F@P|o:

313/16

F_D|o:

735/44

F@`~o:

1241/84

F@`zw:

233/12

F@Q}w:

2162/135

F@P|w:

943/52

F_D|w:

1085/68

F@`~w:

1755/124

F@P~o:

F@P~w:

97/6

F@R~o:

129/10

F@R~w:

25/2

FHQSO:

comment: This is the cycle $C_7$.

FGSkg:

76/3

FCXPW:

344/11

Fk?gw:

406/17

FK_qW:

434/19

FGC{o:

139/4

FICkW:

FKHGw:

346/11

F?L\_:

314/11

FGK]G:

180/7

F_StW:

804/41

FGUPw:

678/29

FOLQw:

620/21

FoCiw:

768/37

FKG]W:

201/10

FAN@w:

674/29

FAYPw:

114/5

F_YPw:

776/39

F@YPw:

80/3

FaG\w:

340/19

FB_mw:

1370/79

F@YRw:

1164/55

F@YVw:

187/12

FGLSW:

97/4

F@Tcw:

104/5

FGLSw:

87/4

F@Tdw:

620/33

F@Tfw:

186/11

F@Umg:

1510/93

F@Xsw:

1304/69

F@hqw:

1246/61

FBJKw:

1474/89

F@Y^_:

405/26

F@Lkw:

334/15

F@h^g:

626/43

F@Y]w:

2308/151

F@Ujw:

2002/111

F@Y^w:

929/68

FHTcw:

278/15

F@Tkw:

1194/61

FHO}w:

1098/65

FALlw:

404/23

F@Tlw:

1006/57

FGL^w:

3296/209

FAdl_:

908/51

F@U^?:

FBebW:

1604/101

F@fbo:

FHQZo:

664/33

FBO|W:

212/11

FHQ[w:

353/20

FHQ^o:

2648/187

FD`jw:

2442/163

F@jRw:

1226/83

FHQZw:

194/11

FHQ^w:

1310/99

F@W}g:

634/33

F@L[w:

229/10

F@T\W:

296/15

F@Lmw:

1004/57

F@MZw:

314/15

FAS|w:

92/5

FAK~W:

2016/115

F@L^w:

82/5

F@Vcw:

212/13

F@U^W:

2318/153

F@UZw:

1844/99

F@U^w:

128/9

FHQ}o:

2806/205

FGN^o:

38/3

FGU|w:

1895/138

FGL}w:

3352/209

FHQ}w:

584/45

FGN^w:

860/71

F@d~o:

3988/299

F@Vlw:

1896/139

F@dzw:

171/10

F@T|w:

618/37

F@U}w:

502/35

FGN\w:

3800/279

FAM~w:

802/63

F@T~o:

63/4

F@T~w:

179/12

F@V~o:

233/20

F@V~w:

45/4

FCX_w:

91/3

FK`_w:

191/6

F_KuW:

65/3

FOTPw:

82/3

F`CmW:

752/33

F`G]w:

56/3

F_Krw:

376/15

F_Kvw:

82/5

F?\t_:

65/3

FK_yo:

223/12

F@\cg:

152/7

F`KqW:

343/12

FCdj_:

214/11

FDhaw:

395/24

FBZ@w:

1364/69

FDUbW:

776/47

F`IZo:

1546/93

FI_xw:

311/15

FaG{w:

349/19

F`H[w:

52/3

FBZDw:

2632/183

FSOzw:

604/39

FI_zw:

2098/111

FB`lw:

2426/159

F`H\w:

604/39

FI_~w:

325/24

FKcqW:

840/41

F@Tl_:

356/15

FAdtO:

808/39

F@LuO:

181/8

F`Maw:

761/44

FDdbW:

1462/79

FJQHw:

329/16

F@nBg:

747/44

F`Oxw:

553/20

FoCyw:

673/36

FJQLw:

1271/84

F`_zw:

146/9

FDPlw:

146/9

FGdrw:

1015/52

F@ptw:

1179/76

FGdvw:

475/34

F@]eg:

774/47

F@X\g:

157/8

F@UuW:

470/27

FHFKw:

710/41

FB_zW:

1250/61

FAgzg:

1192/55

FEHkw:

1402/79

F`Dkw:

1456/85

FAK|W:

239/10

FCLZW:

190/9

F@S}W:

1136/55

FGL[w:

41/2

F_YXw:

1334/75

F@huw:

1216/81

FDO~W:

2422/159

FCLjw:

1928/99

FAMjw:

485/26

F@qZw:

2236/141

F_S|w:

2234/139

FGc~w:

3624/257

F`KuW:

239/12

F`Gyw:

386/15

F`FHw:

146/7

FOLYw:

132/5

F_oxw:

311/15

F`G}w:

256/15

F_Kzw:

362/15

F_K}w:

266/15

F_K~w:

232/15

F@Neo:

65/4

F@NMg:

348/19

F@L^G:

313/16

FB`kw:

725/42

F@ozg:

597/28

F@K}W:

473/20

F@NNg:

303/20

F@o~g:

303/20

F@o}w:

1109/68

F@NJw:

967/52

F@o~w:

113/8

FK`zo:

43/3

FK`~o:

749/60

FK`zw:

55/4

FK`~w:

787/66

FI`|o:

2692/183

FC\ng:

464/35

FI`|w:

55/4

FCXzw:

179/10

FGezw:

3684/257

F_L|w:

3682/255

FCX~w:

5876/465

FGd~_:

227/16

FG]^g:

1047/80

FGd~g:

1991/148

FGdzw:

203/12

F@p|w:

1849/132

FGd~w:

2945/236

F_N~o:

683/60

F_N~w:

659/60

FEOhW:

155/4

FGW[g:

104/3

F?K}_:

139/4

FoCZW:

93/4

FWCYw:

71/2

F_opw:

343/12

FAU`w:

313/12

F@N@w:

63/2

FoCZw:

FEGmw:

F@NBw:

137/6

F@NFw:

67/4

F@K}w:

104/5

F@K~w:

96/5

F@N^o:

133/10

F@N]w:

437/30

F@N^W:

946/69

F@L}w:

3084/185

F@N^w:

2803/220

F@L~o:

163/10

F@L|w:

98/5

F@L~w:

232/15

F@N~o:

61/5

F@N~w:

59/5

FBY^G:

946/69

FIK}W:

256/15

FIo|g:

701/51

F@\sw:

1061/60

FC\rW:

1109/60

FCdzo:

303/20

FAY|o:

1317/92

FINLw:

268/21

FBYZw:

65/4

FAmrw:

1991/148

FC\vW:

268/21

FBY^w:

787/66

FBY^?:

584/39

FLr@w:

3020/231

FF`jW:

2738/195

FAM~O:

2598/181

F_L|o:

2506/165

F@N^O:

72/5

Fk_zw:

1135/93

FC^bw:

4222/327

FGnRw:

2116/165

F_]vw:

2218/195

F_|tg:

4762/399

FFo~W:

2398/219

FWN]w:

191/17

FC^vW:

191/17

FENnw:

2501/240

FBd~W:

6416/537

FC\zw:

47/3

FAmzw:

2857/222

FC\~W:

341/28

FC\~w:

499/44

FBNmw:

3214/271

F@t~g:

6438/545

F@\}w:

2551/168

F@]}w:

5728/451

F@^^w:

8996/799

FG]}w:

3070/253

F_]~w:

9584/897

F@v~w:

1984/201

FBejW:

2574/175

FEXhw:

2166/115

FELlW:

2524/171

FGlug:

FBUlW:

621/40

FIS|W:

1128/65

FBMmW:

2516/169

FJQ\W:

272/19

FAd|o:

2376/155

FAizo:

1246/81

F@L}o:

2066/115

FG[}g:

1126/65

F@h}o:

1290/89

FEW~W:

1052/81

FHNMw:

4198/321

FClrw:

3892/281

FA]rw:

3442/209

F@nRw:

3914/289

FG^Tw:

1052/81

FA]vw:

1549/128

FG\sw:

35/2

FIS|w:

236/15

FA\tw:

883/56

F@\tw:

327/20

F@\vw:

293/20

FHL[w:

2076/115

F@T|o:

344/19

F@]uW:

179/12

F@L|o:

322/15

FHM]w:

3904/285

F@]uw:

558/41

F@]rw:

3174/185

F@]vw:

826/65

F@^vo:

2248/195

FHN]w:

119/10

F@l~g:

62/5

F@^vw:

3134/285

F@\~g:

149/10

F@\|w:

473/30

F@\~w:

211/15

F@]~w:

8436/715

F@^~w:

52/5

F@~vg:

599/60

F@~uw:

10322/1005

F@^~o:

54/5

F@~vw:

142/15

F@~~w:

136/15

FBY~o:

3371/292

FDhzw:

751/60

FBY}w:

1609/136

FBX|w:

179/12

FHU}w:

6444/545

FBY~w:

4705/428

FI\tw:

44/3

FBX~w:

83/6

FBZ~o:

317/30

FBZ~w:

61/6

F?]u_:

FWD[o:

81/4

FGc}_:

432/23

FBj@w:

310/19

F`YPw:

1492/87

FIe`w:

65/4

FIebw:

643/44

FIefw:

129/10

comment: This is the wheel $W_7$.

FHo}g:

2780/199

FM`hw:

2722/191

Fie`w:

53/4

FKK}W:

514/35

F`Lkw:

2482/159

FKL\W:

645/44

FY_}w:

1131/92

FKNJw:

1049/80

FKNNw:

23/2

FHU^G:

2738/195

FKWyw:

1082/57

FIMmw:

2108/163

FHd\w:

977/71

FKL^W:

4206/323

FHUZw:

3448/209

FHU^w:

2068/171

FJQ\O:

170/11

FHU^?:

236/15

Fbj@w:

2878/209

FDprW:

72/5

FIdtW:

2638/177

FIc~G:

1366/97

FIM\W:

Fh_}w:

4434/353

FSTjw:

147/11

FDZJw:

4128/313

FIejw:

147/11

FoL^w:

233/20

Fbc~W:

7076/633

FQdzw:

1577/129

FKNmw:

6748/585

FQL}w:

6320/521

FHd}w:

1577/129

FoNZw:

6770/593

FaM~w:

85/8

FB]vW:

3308/279

FBhzw:

2623/168

FBh|w:

1471/114

FBh~w:

9258/823

FB]uW:

4334/335

FBY}o:

1107/88

FDp~o:

3544/319

FBh}w:

6322/523

FDpzw:

3167/264

FBj^w:

9874/935

FBj~o:

10628/1045

FBj~w:

2042/209

Fo\sw:

2411/204

FIU|o:

749/60

FBY|o:

131/10

FbY\w:

899/82

FImrw:

35/3

FImvw:

163/16

FQ\sw:

2203/172

FJY[w:

67/5

FB\tW:

907/56

F`Lzo:

351/20

FXT[w:

1089/85

F@\~_:

242/15

FB]^G:

4408/345

FJY\w:

3301/276

F`\tw:

414/35

FJYZw:

919/60

FB^dw:

414/35

FJY^w:

1209/110

FT\uW:

3701/340

FI}vg:

5557/564

Fdhzw:

1333/132

FInvw:

1921/206

FLh}w:

5321/524

FS\zw:

133/12

FIm~w:

461/48

FIn~w:

1513/172

FIl~g:

557/52

FI]|w:

133/12

FJY}w:

2501/232

FI]~w:

2321/228

FI\|w:

169/12

FB\|w:

43/3

FB\~w:

53/4

FD\~W:

4801/438

FB]~W:

9590/871

FB]|w:

179/15

FB]~w:

6683/642

FB^~w:

115/12

FkUhw:

1191/100

FsLZW:

FHU}o:

4446/355

Fo^Pw:

2340/193

F@]~_:

4136/315

FA]~_:

740/59

FpUZw:

254/23

Fbh\w:

7102/641

FBnbw:

6652/569

FFYmw:

254/23

FBnfw:

41/4

FDx~g:

1333/132

FPvZw:

5107/492

FBnvW:

10654/1053

FHu~w:

14766/1543

FBy}w:

5108/493

FB]}w:

4589/408

FBn^w:

834/85

FBn~w:

2914/323

FF^nW:

55/6

FI~tw:

7909/860

FB^~o:

599/60

FBn~o:

15216/1615

FB~vw:

173/20

FB~~w:

33/4

F@~v_:

317/30

FbY|o:

235/22

FFzbw:

39/4

FFzfw:

FK~v_:

289/30

F]p|w:

2057/230

Fs\zw:

55/6

Fs\~w:

101/12

FFz~o:

83/10

FFz~w:

79/10

FF~~w:

15/2

FGeZ_:

361/12

FJaHw:

43/2

F`N@w:

161/6

FJaJw:

73/4

FJaNw:

FaK|W:

2432/155

FPLYw:

99/5

FKcyw:

82/5

FEXlw:

4102/305

FHeZw:

946/65

FPTZw:

3324/185

FKW}w:

1360/99

F`MZw:

946/65

FPT^w:

1513/120

FJYKg:

779/44

FJ_}W:

1303/84

FKdjg:

1259/76

FjaHw:

29/2

comment: This is the Moser spindle.

FKLkw:

599/36

F`K}W:

1139/60

FwC}w:

67/5

FKYZw:

495/34

F`NNw:

99/8

F`Kyw:

124/5

F`K}w:

242/15

F`Kzw:

116/5

F`K~w:

218/15

FeK~W:

231/20

FKdzw:

769/60

FQT|w:

769/60

F`N^W:

3301/276

F`N^w:

1209/110

F`L~o:

1294/105

F`Lzw:

509/30

F`L|w:

1148/85

F`L~w:

1814/155

F`N~o:

209/20

F`N~w:

201/20

FKzPw:

4648/377

FQT|o:

4202/305

FI]\g:

4320/329

FwL[w:

240/19

FB]lg:

167/12

Fbo|w:

1739/152

FJejw:

1621/132

FJenw:

317/30

Fkoxw:

4674/385

FpLYw:

852/65

FkYXw:

1145/92

FPT}o:

2174/169

F`L|o:

784/55

F`v`w:

253/20

FpL]w:

1747/155

F`]rw:

6528/545

Fbg}w:

6976/615

F`]vw:

2036/195

FJ]^G:

1362/115

FL]uW:

2436/221

FJnNg:

3656/365

FTpzw:

2627/255

FJq~w:

759/80

FJd~W:

9858/895

FK\zw:

221/15

FK\|w:

785/69

F`\|w:

4723/420

FK\~w:

6869/660

FTX}w:

10496/1015

FK]~w:

3641/372

F`l~g:

2627/255

F`t|w:

10046/943

F`]~w:

14578/1495

FK^~w:

2994/335

Ftpzw:

181/20

FK^~o:

3128/335

FK~vw:

128/15

FK~~w:

122/15

FJY}o:

1739/152

FJvdw:

5494/551

FLpzw:

4951/456

FLp|w:

10528/1027

Fbh|w:

2635/258

FLp~w:

15216/1615

FLr~o:

858/95

FLr~w:

164/19

FBnvO:

7434/689

F`]~_:

7316/665

Fb]lg:

131/12

FLvbw:

2749/276

FLvfw:

55/6

FLvvO:

126/13

Flp|w:

16310/1807

FM^lw:

15672/1687

Fk]~w:

5609/660

FJn^W:

270/29

FJ]}w:

7097/696

FJn^w:

21596/2465

Fbn~w:

4392/551

FZn]w:

11537/1380

FLv~o:

23062/2755

FL~vw:

902/115

FL~~w:

856/115

FU\~W:

1307/141

Fb]|w:

314/33

FLl}w:

7523/786

FR^^w:

10805/1236

FJm~w:

433/48

FJn~w:

2119/258

FR\}w:

395/39

FJ]|w:

21/2

FJ]~w:

547/57

FJ\|w:

27/2

FJ\~w:

38/3

FJ^~w:

FJ~~w:

23/3

FN~~w:

83/12

FJn~o:

11111/1290

FJ^~o:

47/5

Fj]|w:

103/12

FJ~vw:

121/15

Fjm~w:

47/6

FNz~w:

439/60

FNz~o:

463/60

F]~vw:

36/5

F]~~w:

34/5

F^~~w:

32/5

F~~~w:

comment: This is the complete graph $K_7$.

Definition

For a connected simple graph $G$, $\mathrm{Kf}(G)=\sum_{\{u,v\}\subset V(G), u\ne v}\Omega_G(u,v)$, where $\Omega_G$ is the resistance distance [1] [2] [3].

Parameters

$G$

— graph (a simple connected graph, named by its canonical graph6 string)

Formulas

(1)

If $G$ has $n$ vertices and nonzero Laplacian eigenvalues $\mu_1,\ldots,\mu_{n-1}$, then $\mathrm{Kf}(G)=n\sum_{i=1}^{n-1}1/\mu_i$ [3].

(2)

If $\lambda(G,x)=x^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x$ is the Laplacian polynomial of a connected graph on $n$ vertices, then $\mathrm{Kf}(G)=-n c_2/c_1$.

(3)

$\mathrm{Kf}(K_n)=n-1$.

(4)

$\mathrm{Kf}(P_n)=(n^3-n)/6$ for $n\geq1$, and $\mathrm{Kf}(C_n)=(n^3-n)/12$ for $n\geq3$.

(5)

$\mathrm{Kf}(K_{1,n-1})=(n-1)^2$ for $n\geq1$.

Comments

(6)

Graphs are named by their graph6 string [4], McKay's compact encoding, taken after a canonical relabelling so that the name depends on the graph and not on how its vertices happen to be numbered. Recover the graph in Sage with Graph(name). If typing a graph6 name directly, quote it as a raw string, for example Graph(r'FC\ng'). Produce the name used here with g.canonical_label(algorithm='sage').graph6_string(). The same names index the characteristic polynomials, the Laplacian polynomials, the chromatic polynomials, and the Tutte polynomials.

(7)

The Laplacian is the combinatorial Laplacian $L(G)=D(G)-A(G)$, not the normalised Laplacian and not the signless Laplacian $D(G)+A(G)$. The resistance distance $\Omega_G(u,v)$ is the effective resistance between $u$ and $v$ in the electrical network obtained by replacing each edge by a unit resistor.

(8)

The Laplacian eigenvalues are ordered $\mu_1(G)\geq\mu_2(G)\geq\cdots\geq\mu_n(G)=0$, with multiplicity.

(9)

Thirty-five entries give a common name for the graph. Every entry is identified by its canonical graph6 string.

Programs

(P1)

Sage

from sage.graphs.graph import Graph
from sage.rings.rational_field import QQ

G = Graph('FjaHw')               # the Moser spindle, by its name here
p = G.laplacian_matrix().charpoly()
n = G.num_verts()
-QQ(n) * QQ(p[2]) / QQ(p[1])

# and the name of a graph you have:
G.canonical_label(algorithm='sage').graph6_string()

References

[1]

D. J. Klein and M. Randic, Resistance distance, Journal of Mathematical Chemistry 12 (1993), 81-95. (doi)

Links

[2]

MathWorld: Kirchhoff Index

[3]

Wikipedia: Resistance distance

[4]

Brendan McKay: graph6 and sparse6 formats

Similar tables

Laplacian polynomials of connected graphs — stores $\lambda(G,x)=\det(xI-L(G))$ for the same graphs under the same names; $\mathrm{Kf}(G)$ is assembled from the two lowest-order nonzero coefficients of $\lambda(G,x)$

Characteristic polynomials of connected graphs — stores the characteristic polynomials $\det(xI-A(G))$ of the same graphs under the same names

Chromatic polynomials of connected graphs — uses the same canonical graph6 index for connected graphs on at most seven vertices

Tutte polynomials of connected graphs — uses the same canonical graph6 index for connected graphs on at most seven vertices

Matching-generating polynomials of connected graphs — uses the same graph names and stores the ordinary generating polynomial for matching counts

Signed matching polynomials of connected graphs — uses the same graph names and stores the signed matching polynomials

Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds every connected graph on at most $7$ vertices, all $996$ of them)

How they were obtained:

The generator enumerates connected simple graphs with $1\leq |V(G)|\leq7$ using Sage, takes each graph's canonical_label(algorithm='sage'), and stores the canonical graph6 string. It computes $L(G)=D(G)-A(G)$ exactly over $\mathbb{Z}$, computes $\lambda(G,x)=\det(xI-L(G))$ exactly, and returns $-n c_2/c_1$ as a rational number.

The build checked that the 996 graph6 strings are exactly the keys of the characteristic polynomial table. On every row, the value from Formula (2) agreed with an exact resistance-distance computation from the inverse of $L+J/n$, and applying the formula to the Laplacian polynomial in that table gave the value in this table. Formulas (3), (4), and (5) agreed for every complete graph, path, cycle, and star in the table.