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{"Title": "Distance Estrada indices of connected graphs", "Definition": "The distance Estrada index of a connected simple graph $G$ is $\\operatorname{DEE}(G)=\\sum_{i=1}^{|V(G)|}e^{\\mu_i}$, where $\\mu_i$ are the eigenvalues of its shortest-path distance matrix, counted with algebraic multiplicity CITE{GungorBozkurt}.", "Parameters": {"g": {"type": "Symbolic", "title": "graph", "display": "$G$", "constraints": "a connected simple graph on at least two vertices, identified by its graph6 string after Sage canonical_label(algorithm='sage')"}}, "Tags": ["combinatorics", "eigenvalue"], "Links": {"GungorBozkurt": {"title": "A. D. G\u00fcng\u00f6r and \u015e. B. Bozkurt, On the Distance Estrada Index of Graphs, Hacettepe Journal of Mathematics and Statistics 38(3) (2009), 277\u2013283, Definition 2.1", "url": "https://dergipark.org.tr/en/download/article-file/86791"}, "Graph6": {"title": "Brendan McKay: graph6 and sparse6 formats", "url": "https://users.cecs.anu.edu.au/~bdm/data/formats.txt"}, "McKayGraphs": {"title": "Brendan McKay: connected graph catalogues", "url": "https://users.cecs.anu.edu.au/~bdm/data/graphs.html"}}, "Data properties": {"type": "R", "rigour": "proven", "digits": "100", "complete": "no", "complete-note": "it holds all 142 connected simple graphs on two through six vertices, with counts 1, 2, 6, 21 and 112 by order. This complete small-order reference census covers trees, cyclic graphs, bipartite and non-bipartite graphs, and regular and irregular distance patterns for identifying shortest-path matrix-exponential calculations. Six vertices is a practical, noncanonical boundary; seven would add 853 graphs, mainly extending the census", "rigour details": "Each distance matrix and characteristic polynomial is computed over the integers. Certified real algebraic roots are converted to outward-rounded intervals, then exponentiated and summed with their algebraic multiplicities in Arb ball arithmetic. The generator uses 96 guard bits beyond the 100-digit target. The minimum measured relative accuracy was 424 bits, exceeding 127 decimal digits; each stored value claims only 100 significant digits.\n\nAn independent checker decodes graph6 directly, computes shortest paths by Floyd-Warshall, and sums $\\operatorname{tr}(\\mathcal D^k)/k!$ using exact integer powers and rational arithmetic. For $R=\\max_v\\sum_u d_G(u,v)$ and $N+2>R$, the tail after degree $N$ is bounded by $nR^{N+1}/((N+1)!(1-R/(N+2)))$. The tail is less than $10^{-125}$ for every graph, with degree at most 187. Zero-matrix, identity-matrix, $K_2$ and $P_3$ controls precede the comparison, which agrees on all 142 entries. Every enclosure is checked for finiteness and narrowness.\n\nThe complete-graph formula agrees on all 5 cases, the complete-bipartite formula on all 9 cases, and the spectral bounds on all 142 graphs. The distance characteristic polynomials agree exactly with HREF{T537}[the stored polynomials], and the largest roots agree with HREF{T536}[the stored distance spectral radii]. Trace and squared-trace checks test the root multiplicities. The canonical graph identities agree with an independent nauty census, the adjacency and Laplacian tables, and McKay's counts. The attached generator, independent checks, argument-selection note and numerical evidence reproduce these results."}, "Display properties": {"number-header": "$\\operatorname{DEE}(G)$", "group parameters": [["g"]]}, "Numbers": {"A_": {"number": "3.086161269630487556955811241514123365203058224731727409474804429421538126098447397928529452871086071", "comment": "This is the complete graph $K_2$."}, "BW": {"number": "15.98062106977042375057082809489108821161682674849701995126753205004485511246927153533782085494150776", "comment": "This is the path $P_3$."}, "Bw": {"number": "8.124814981273534870421475000897929548071937832615382993102801425917496787568857370098607033770961082", "comment": "This is the complete graph $K_3$."}, "CF": {"number": "104.9365181929889927958468098276271895456062108959116811150647094128856405638320476321899306007797364", "comment": "This is the star $K_{1,3}$."}, "CL": {"number": "175.4639383067344962489316396768200439766588421227751903581091971793356896222334257071466066083257718", "comment": "This is the path $P_4$."}, "CN": "61.26764791115747562437093605785095423836743255627883924506306041940296394947161538544619125696395942", "C]": {"number": "55.86882059961746446189826019280584720960600013043322048876291170386180884813884486155117680692604824", "comment": "This is the cycle $C_4$."}, "C^": {"number": "36.29137891902769446831171514050725280098901944475375545382649094874239887324372088560526396015941124", "comment": "This is the diamond graph."}, "C~": {"number": "21.18917524670199470571510096506610049932534123164945364790244463479123036532667314127593898329077663", "comment": "This is the complete graph $K_4$."}, "D?{": {"number": "740.1390320287034460941666742634856479754055314406149312736876415936413332097925535820224483453343557", "comment": "This is the star $K_{1,4}$."}, "D@s": "1737.016217792652681289673509796859786210824754652700507338010362358439844699853470605450134741199440", "D@{": "482.3844509738899418167922560072862102922130421921294971676288513318215177953855466691478111518938923", "DBg": {"number": "3977.953213670659586321467988963604676167625668603249362080363168448001539195829333034652649127799788", "comment": "This is the path $P_5$."}, "DBk": {"number": "764.3874258021335471482632729916093816196706546878658052860923592038629021977201236038796689662790250", "comment": "This is the bull graph."}, "DB{": {"number": "318.4669443907120633829313084362739363583900033604748376592217693065320309025444115175466999989884363", "comment": "This is the dart graph."}, "DFw": {"number": "284.9172775131309382452539683134928077743288113490558022256235434463147672743382716195972018852226036", "comment": "This is the complete bipartite graph $K_{2,3}$."}, "DF{": "216.6813092555594938464563578062285417005815340219033543443346543937305083582572811275513530049695082", "DIk": "692.5014592056168161300806647509637089909758032505931131687566683751517350935367613979709891153244505", "DJk": "502.0777202792142142584341708502316217887720171868026073372025431953093050837653425208314557476840570", "DJ{": "210.6687426668196416571370416438388408090102726982607937713490145147155996839658256428694859680460854", "DK[": "1107.369635575373552471662235349859248585313725488643779987246534396341523562542326574162667387539143", "DK{": {"number": "300.6159883030068430320958689230189405329492637655407420257375572672802600282962350588964722011646549", "comment": "This is the butterfly graph."}, "DLo": {"number": "404.9397222639734216223495085158231868552988342954011977100617883191589791965711910616296268597024564", "comment": "This is the cycle $C_5$."}, "DL{": "200.3050661192779489996401079213728423584745087455161663090305134108452956533475664426514121118943097", "DNw": "187.4191441454300667414668476377139644516972241266757911721477065561184677097738818775466742878979386", "DN{": {"number": "135.9582449210763215791408893727953893470174868198435454946870686773275869252899010126976869839237242", "comment": "This is the house X graph."}, "D]{": {"number": "126.7215804776497176194648670288342060459341383501232369880838744728710754363647079646363012061165675", "comment": "This is the wheel $W_5$."}, "D^{": "87.09231113578144277003639168415013602479535217064748588958946915426179123470424787183537921855454236", "Dbk": {"number": "281.6974367870896967660724570224676227278070584281452065771240738014822221542977056925487453888023353", "comment": "This is the house graph."}, "D~{": {"number": "56.06966779783000836449235628350672187257398156274114006385794116534350808291546869510556965932841272", "comment": "This is the complete graph $K_5$."}, "E?Bw": {"number": "5338.936272056536891806300916033921698439467102570399075278023517751140773223659017855425238402206889", "comment": "This is the star $K_{1,5}$."}, "E?Fg": "15839.94846300674487695908782920188892111507321506188410414957484462809685296107621971163182679158140", "E?Fw": "3752.026112850319582145824305168195531993772868534718502308873500633210548244553712542938656762600955", "E?NG": "22027.75982790624056429695148312012926417494652573148518178589720463102121862364581332942138999669190", "E?NW": "8031.048227455407433164623527485809584700603556170441649678539650145905359717635270328260364009442815", "E?Nw": "2668.640414147000215857723461783426918049955083488841560542025899840050326276928886960865765562552216", "E?]o": "15630.34098506027157671493794930072156654980988224243102373739970459511988547322714085370493569964684", "E?]w": "3978.206966829842914536652041988731229378536632927661511306442287879912916203308513098386853229885741", "E?^o": "3454.473365218972049445269284936300423320284180537302411603107998779336036148015878777099918439945507", "E?^w": "1935.393964830938006121947214178882898358511499071085496893642360561586936885391709410505179946500051", "E?~o": {"number": "1746.538126361112822663672666439278093627493408449610508480191167786144052403858508312316207617326369", "comment": "This is the complete bipartite graph $K_{2,4}$."}, "E?~w": "1445.165200253118182493252824872633321828779859896151662599559953318106663779758918748533524753097705", "E@FG": "63500.77439677117064129840522527701731548418049718152809646168098738504107226056481479455636862748182", "E@JW": "7326.190941961713424633983793288498396124990989756029428565159557652462582580074395305907436819056805", "E@NW": "5642.192859588359720943739309031800138204026734948087843666656708712218175054462937487486891990292191", "E@Nw": "1910.893218119807050673341365497338255011462672436922097600081242364288864342700444270708695243598402", "E@QW": "46278.61132461993734097323232875393810595323057583854714826949311540375671784077079378257855211538470", "E@Rw": "2573.954424077091662774568342479748591723121976865398714599409039606328450228499733753189754290462612", "E@UW": "11911.80036306997335287637681705699391103940098131755796064440190282075275391372297985516384026255106", "E@Vw": "1850.228615972263651385651457063086055043031621841498796816158069657512172751373849291536968339556549", "E@YO": {"number": "181556.1102617632977481881539720419638994634975869017354261721686537504891742638798118416766279361030", "comment": "This is the path $P_6$."}, "E@YW": "10517.04417838890892522238796837218649376434839440844126758822435628993890739282436185112735430596333", "E@]o": "12388.95164604071043860567560493486679311350947057101773778928675043782581223878581183457318158987116", "E@]w": "2916.655620037573065275622034384779997956963167807190115889323435993596160647998676167085576602628283", "E@^W": "2739.952737706901738673724917328910093850842499898863431293630878102557692342616654244411239349646587", "E@^o": "2534.639615845742776071575179008836342446331053587005733908399604428627084423234113508273673289699601", "E@^w": "1348.632665381208812107022767457340191430094572633709793469217721351637095820065792151519581105508884", "E@`w": "11109.28957359692569569816913843641836870729755211921551399800027249519722719999857716183416294230012", "E@ow": "14897.25766209238664366851496773833006905926631068082316638303706632230295267020325326629054895260645", "E@vw": "1307.055224891452829405381207631450628116653819840863793839677932456853461274756637367533522041013599", "E@~o": "1209.323337144857367373347227836983441415941931056710752559815073288646445938888870418287184327002483", "E@~w": "979.9749118956918780836361287130042352187777524355235718072842726476609003595527802700579121093996350", "EAMw": "5787.816128320897501443804475177308672467456599095525349367606935785771417065163855661989013712784550", "EA]o": "5034.678625905491002206332037328484935211883826515398061997423878050091030892612100029183962743673201", "EBYW": "5275.358694772996561803839131505554328040891550211224570755072496736377134838592150155877163512748618", "EBYw": "2635.140913962500057058021829829716326092242159344445256012332899342206308024656889272385026569495095", "EBZw": "1294.929820433222525616093711116456839706471341339170453850577036724525505501280651237598221845643896", "EB]w": "2047.279677120054326946663928925332111163979920419610240460282280823118299907549767963329235987265221", "EB^w": "955.1173957580200950282497523277857896543396471517409861023229202390808815830391724542716171916922013", "EBhw": "3915.470339081033526755628220802569350446262533328229994433609007999408468491593010070211347773753071", "EBjW": "1638.227643577834922590840158161131741905108621645369609375614917013584042830871300731860081287541737", "EBjw": "1249.031253409427948814906460445039379430255979495841929836789046310191772400863152783912457964797616", "EBnW": "1242.132551345918150269003717538909244488275474284112976868201221956106676898199763409950540743084897", "EBn_": "1573.151856980934808919827158975331388368031961646593906882105843907290706333260795744513417599534490", "EBnw": "925.9261727370066008083104968131578336403475961481875936618481207110556397789564439640770014644759016", "EB~o": "851.2248620521815665450487666798153681310685833310608077482570549459101780968341502869034045105782828", "EB~w": "674.2033558800885599553827237426074770307353315197862522519606139966752333374102153499882056230954189", "ECXw": "7860.814461700763010907670357264677307020052200392370242574109573605282306830947419029778544241067207", "EC\\w": "4261.146889713286735485934396837653296226145100471708951399038847381878723022684441131514353660337322", "EENg": "1647.670076196843591079886655347185928334628256216257430992928774239812637189803630589856487331806659", "EFz_": {"number": "1099.892781389864095266656523739364032553606908110951406883578843894462139457992627376798343455737639", "comment": "This is the complete bipartite graph $K_{3,3}$."}, "EFzw": "615.6118341186009504580938718728425710397654568964473982691707829252919004112669533903349238704590456", "EF~w": "476.3642465987234945563231638263518860758608900552076651272769647178493319672691889504088295477171517", "EGNW": "3578.347557759405590001110027654344679278921995460311663945916242627160778230662226850137525491049118", "EGcw": "23268.21685147475775195422285761911637263347756165707185158628963210667744058937682003995899768664457", "EGdo": "42958.93167856354008660459636029963328875114164340547115022550754600732759352597754808044216177016680", "EGdw": 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"884.5548329535525208705932282614865850941907083446246047244568080417889399586397546298462846388456355", "EK`w": "4901.751024261227629505635857803858610012909159950920886912191405703790890154329193657442692331798638", "EK~o": "815.9445242990311448635842129434337636267779290382998730486129734054634575805024451945699318557614130", "EK~w": "643.4697470287548356137380118018205775757376711832277643141483812003121076539003746479826626189528187", "ELpw": "1140.297135948647527155729844577470412012132312003753985821854934791570976268915460139589013815224973", "ELrw": {"number": "848.9505716009353782179697958747051288244005173908741250146171586516586921352419961425168396340444203", "comment": "This is the wheel $W_6$."}, "ELv_": "1098.953616063299688590487579693716463025666097517620718761243186249035976747166303952506819276467791", "EL~o": "583.2820762342502749771922080292858678188268049623847369952639568518428254697669362447056517526231509", "EL~w": "448.3000081001955380186845037111542505673857440565708369616863418302710380663520524717160276119049165", "ENzw": "425.4409137823095588973010963534599381758693724713757568390306206187356621174452394984179501585495938", "EN~w": "318.1590442413553749833209853949898667185920156400015212021415890930147868750839117884980630406974427", "EPTW": "16260.23864817584769976344963351101003416777084007815319117604948964144421767916683441505930905122227", "ER^W": "841.1000247098061043845877594480410857653433008387522376734954586664561454792909601126126324735992952", "E]~o": "405.8347993424449606840691790283057328161227919948579302583716649431134007562979613749901810186846359", "E]~w": "301.4689923599407621513090017271523866956870161341397027385984104832848846130241110852594669275634161", "E^~w": "217.2817328546657657977534058515789790320655247380573141318821691244713579756455930443285494311706904", "E_Nw": "1785.126955451363729732622343782499185989922565393790483061438845548978809186024070488790157323553054", "E_]o": "2199.822201729650921560542878864119855120864860297005973014487525196123885683180160800996992701733966", "E_]w": "1707.612530876357897210212841369570672844980268865561297953433335651271642605028551133038294087648301", "E`Lw": "5586.926756369160120158257769474816773880142986590601205752812500883446162841609905664335652125447105", "E`NG": "9860.741321840017072906209578700693775979999562797753574955491779683615564650740650437618607646134838", "E`NW": "3705.027998501286553819943892339147047497754099113954794566055296814798522073279459056554202628540837", "E`Nw": "1243.783069204762823491119745692047075409478099841912427082850211225033431875572109500275077529418712", "E`]o": "1566.516976036622516881119928352348587289192424654051656424011388117825616990130463616337619255948667", "E`]w": "1183.614173638173284405231042125891833881285873998829741198101046762410968181333144832375290765436724", "EaMw": "2453.212319164162917296592061668622055482017310728099945978312282128402633201860023187841286403403226", "Ebnw": "610.8061425422281707776824660014152667343495332382607980497762590002213155762641747556638593811923887", "Ejmw": "580.0429259146185535890306822355353344551765027317196697815497162514257071981339605060444192150749935", "Ek]w": "810.8780388757483514896305222418515627954503225198815740973964167812281301604737535706897129646988731", "EoLW": "3212.036923481435148783759837995323370012518996360144104058832186527141780243257704933991344198173252", "Es\\w": "811.9653566535364688218387157993962514770910323822106635269255977860135044886151978428510306755593887", "E~~w": {"number": "150.2525563084338150290931988913595839607167232490378282192920291193993722996830750964741458316286194", "comment": "This is the complete graph $K_6$."}}, "Comments": {"comment-matrix": "The graph is finite, undirected and unweighted. With $n=|V(G)|$, its distance matrix is $\\mathcal D(G)=(d_G(u,v))_{u,v\\in V(G)}$, where $d_G(u,v)$ counts edges in a shortest path and $d_G(u,u)=0$.", "comment-index": "Graphs are identified by their graph6 strings CITE{Graph6} after `G.canonical_label(algorithm='sage')`. Recover a graph with `Graph(key, format='graph6')`; use a raw string literal for a key containing a backslash. These identities agree with HREF{T237}[the adjacency characteristic polynomials] and HREF{T238}[the Laplacian polynomials]. Entries are ordered by vertex count and then by graph6 string.", "comment-spectrum": "The eigenvalues in the exponential sum are the signed roots of the HREF{T537}[distance characteristic polynomial], with their multiplicities. Their largest member is the HREF{T536}[distance spectral radius]. The index uses the unscaled distance matrix and is neither centered nor divided by the number of vertices.", "comment-small": "The one-vertex graph has distance Estrada index $1$. For a wheel $W_n$, $n$ is the total number of vertices, including its hub.", "comment-census": "The selection is the complete connected-graph census on two through six vertices CITE{McKayGraphs}. Common graph names annotate 26 entries; the other graphs are identified by their graph6 strings."}, "Formulas": {"formula-trace": "$\\operatorname{DEE}(G)=\\operatorname{tr}(\\exp(\\mathcal D(G)))=\\sum_{k=0}^{\\infty}\\operatorname{tr}(\\mathcal D(G)^k)/k!$, where $\\mathcal D(G)^0=I_n$ CITE{GungorBozkurt} CITE{EstradaSurvey}.", "formula-complete": "For the complete graph $K_n$, $\\operatorname{DEE}(K_n)=e^{n-1}+(n-1)e^{-1}$, since its distance eigenvalues are $n-1$ once and $-1$ with multiplicity $n-1$.", "formula-bipartite": "For the complete bipartite graph $K_{a,b}$ with $a,b\\geq1$, put $s=\\sqrt{a^2-ab+b^2}$. Then $\\operatorname{DEE}(K_{a,b})=e^{a+b-2+s}+e^{a+b-2-s}+(a+b-2)e^{-2}$. The vectors constant on each part give the quotient matrix $\\begin{pmatrix}2(a-1)&b\\\\a&2(b-1)\\end{pmatrix}$; vectors summing to zero within a part have eigenvalue $-2$.", "formula-bounds": "For a connected graph on $n\\geq2$ vertices, $\\operatorname{DEE}(G)>n$ and $e^{\\rho_{\\mathcal D}(G)}<\\operatorname{DEE}(G)<n e^{\\rho_{\\mathcal D}(G)}$, where $\\rho_{\\mathcal D}(G)$ is the HREF{T536}[distance spectral radius]. The first inequality follows from strict convexity of the exponential and $\\operatorname{tr}\\mathcal D(G)=0$."}, "References": {"EstradaSurvey": {"bib": "Ernesto Estrada, The many facets of the Estrada indices of graphs and networks, SeMA Journal 79 (2022), 57\u2013125, Definition 23 and equation (16.6).", "doi": "10.1007/s40324-021-00275-w"}}, "Similar tables": [{"table": "HREF{T537}[Distance characteristic polynomials of connected graphs]", "relation": "its roots, counted with multiplicity, are the exponents in the distance Estrada index"}, {"table": "HREF{T536}[Distance spectral radii of connected graphs]", "relation": "stores the largest distance eigenvalue for the same graphs"}, {"table": "HREF{T237}[Characteristic polynomials of connected graphs]", "relation": "uses the adjacency matrix with the same canonical graph6 identities"}, {"table": "HREF{T238}[Laplacian polynomials of connected graphs]", "relation": "uses the combinatorial Laplacian with the same canonical graph6 identities"}], "Programs": {"program-sage": {"language": "Sage", "code": "import numberdb.sage\nfrom sage.graphs.graph import Graph\nfrom sage.rings.qqbar import AA\nfrom sage.rings.real_arb import RealBallField\nfrom sage.rings.real_mpfi import RealIntervalField\n\nG = Graph('CL', format='graph6') # the path P4\np = G.distance_matrix().charpoly('x')\nB = RealBallField(430)\nI = RealIntervalField(430)\nsum((B(I(r)).exp() * m for r, m in p.roots(ring=AA)), B(0))\n# Stable graph identity:\nG.canonical_label(algorithm='sage').graph6_string()"}}, "Keywords": ["distance Estrada index", "DEE", "distance matrix exponential", "distance spectrum", "shortest path", "spectral graph theory", "complete graph", "complete bipartite graph", "path graph", "cycle graph", "star graph", "wheel graph", "diamond graph", "bull graph", "dart graph", "butterfly graph", "house graph", "house X graph"]}