Maximum Walk Entropy Implies Walk Regularity

Transcription

1 Maxmum Walk Etropy Imples Walk Regularty Eresto Estraa, a José. e la Peña Departmet of Mathematcs a Statstcs, Uversty of Strathclye, Glasgow G XH, U.K., CIMT, Guaajuato, Mexco BSTRCT: The oto of walk etropy S, G for a graph G at the verse temperature was put forwar recetly by Estraa et al. (4) [6]. It was further prove by Bez [] that a graph s walk-regular f a oly f ts walk etropy s maxmum for all temperatures. Bez (4) [] cojecture that walk regularty ca be characterze by the walk etropy f a oly f there s a, such that S, f a oly f the S G, l. We also prove that f the graph s regular but ot walk- regular G s maxmum. Here we prove that a graph s walk regular S G, l for every graph s ot regular the a lm S G, l lm S G, S G, l for every for some.. If the MSC: 5C5; 56; 8C Keywors: Walk-regularty; Graph etropes; Graph walks. Itroucto. The cocept of walk etropy was recetly propose as a way of characterzg graphs usg statstcal mechacs cocepts [6]. For a smple, urecte graph G, E oes a ajacecy matrx the walk etropy s efe as S G, p l p, wth

2 where p e a Z ( k B s the Boltzma costat a T the absolute kt B temperature). Here Z Tr e represets the partto fucto of the graph, frequetly referre the lterature as the Estraa ex of the graph [3, 4, 9]. The term e represets the weghte cotrbuto of every subgraph to the cetralty of the correspog oe, kow as the subgraph cetralty SC of the oe [7, 5, 8]. The walk etropy calle mmeately the atteto the lterature [] ue to ts may terestg mathematcal propertes as well as ts potetals for characterzg graphs a etworks. I [6] the authors state a cojecture whch was subsequetly prove by Bez [] as the followg Theorem.. [] graph s walk-regular f a oly f S G, l for all. Bez [] also reformulate aother cojecture state by Estraa et al. [6] the followg stroger form Cojecture.. [] graph s walk-regular f a oly f there exsts a such that S G, l. followg thr cojecture to be cosere here was orgally state by Estraa et al. [6] as the Cojecture.3. Let G be a o-regular graph, the S G, l for every. I ths ote we prove these two cojectures, whch mmeately mply that the walketropy s a strog characterzato of the walk-regularty graphs a also gves strog mathematcal support to the stregth of ths graph varat for stuyg the structure of graphs a etworks.. Ma results We start here by statg the two ma results of ths work.

3 Theorem.. Let be the ajacecy matrx of a coecte graph G. The the followg cotos are equvalet: (a) G s walk-regular; (b) (c) () k ; k has a costat agoal for atural umbers e has costat agoal e has costat agoal for ; S G, l. (e) The walk etropy Theorem.. Let be the ajacecy matrx of a graph G. The oe of the followg cotos hols: S (a) G s walk-regular. The G, l for every (b) G s a regular but ot walk-regular graph. The Moreover, lm S G, l lm S G (c) There s some ;, such that S G, l for every. S G, l for every. 3. Proof of the Theorem We start by seeg that (a) clearly mples (b). For (b) mples (a), let T p T p p T be the characterstc polyomal of the graph G. The Cayley-Hamlto theorem yels If k has a costat agoal for atural umbers p p m m m has a costat agoal. k m a m, the p. 3

4 Clearly, (a) mples () whch s equvalet to (c). We shall prove that () mples (b). We follow the techques use for Theorem. []. For, we coser Tre e to be a real aalytc fucto. s power seres k! 3! 4! usg that G has o loops a fucto 3 4 k 3 a the lmt k k k s the egree of the oe Coser the aalytc lm s epeet of the oe, showg that G s regular. Repeatg the argumet we get successvely that k 3,4. k s epeet of the oe for () mples (e): let y be the costat value of the etres of e. The Z y a S y y G, l l. y y (e) mples (a): follows from Theorem.. Q.E.D. 4. uxlary eftos a results Before statg the proof of the Theorem. we ee to trouce some eftos a auxlary results, whch are gve below. We rem the reaer that gve a set X x x of real umbers, the varace s efe as,, s 4

5 X EX EX s s x x. s s Defto 4.: Gve a matrx M wth agoal etres the agoal varace as M,,M, ot all zero, we trouce M M,, M. M Let us ow state a proof the followg auxlary result. Proposto 4.: Let be the ajacecy matrx of a coecte graph G. The oe of the followg cotos hols: (a) e has costat agoal (b) e has o costat agoal etres a G s a regular graph. The e a lm e ; for (c) There s some such that e for every. Proof: We stgush the followg mutually exclug cases accorg to Theorem : () G s walk-regular, equvaletly, () e has ot costat agoal, for ay e has costat agoal.. The e for. Observe that for lm e e lm Z we have a e, where the (Perro) egevector of correspog to the maxmal egevalue. I that stuato e e. Z lm : : Therefore lm e s equvalet to beg costat, or G beg regular. s 5

6 If G s ot regular the the aalytc fucto e Clearly, there s some such that e for for every. Q.E.D. a lm e. We cotue ow wth some other auxlary results ee to prove the Theorem. Let be the egevalues of, such that etres y y y have,, of j j e we efe a vector z y y y. For the vector of agoal l l,,l of real umbers. We z ze y l y wth z l y l et e, where the equalty s a rect applcato of Haamar s theorem for the postve efte matrx Grgesoh [] states that Theorem 4.3. Let c,3, 4 a c e / 5 The [], e. The remarkable result of Borwe a a let z be efe as before. c z z ze. Remarks 4.5. (a) Observe that a pror t s ot eve clear that the sum z ze s postve. (b) Borwe-Grgesoh equalty mproves a prevous bou gve by Kostat a Mchor []. 5. Proof of the Theorem We kow that S G, l for every. Observe that for Z Tr e a the vertex etropy s y y S G Z y y Z z e z, l l l l Z Z Z Z 6

7 The Borwe-Grgersoh [] equalty yels c S G Z z Z, l Moreover, the arthmetc mea-geometrc mea equalty yels / / Tr Z Tr e y e e We stgush two stuatos at : () z, that s y for,,. The, Z Tr e y a therefore S G, l Z l. Z I partcular, for ay, the arthmetc-geometrc mea equalty yels / / Tr Z Tr e e e e whch mples that all e have the same value, that s that all have the same value. Tr, we have that for,,. The, the graph G s empty (t has o Sce lks) a S G, l for ay. () z. The there s a fferetable fucto c such that S G, l Z z l. Z 7

8 Sce Z there s a fferetable fucto e satsfyg e such that e. S G, l z such For every M, usg the compactess of the terval, M, there exsts a M that e for, M S G, l.. Moreover, recall from [3] that Ths lmt s l except whe there s a commo value c,,. The latter property mples that G s a regular graph. We coser these cases separately. (3) ssume that G s ot a regular graph. The such that for, we have S G, l. Therefore there exsts a e z. S G, l. that s, (4) ssume G s a regular graph. We may assume that G s ot walk-regular. The, accorg wth the aalyss [3], the maxmal value lm S G, l lm S G,. Q.E.D. I closg, the maxmum of the walk etropy at for the walk-regular graphs. Ths meas that, walk-regularty graphs. Refereces: S G, l s ot attae for ay Moreover,,.e., S G, l, s attae oly S G ca be use as a varat to characterze 8

9 [] M. Bez, ote o walk etropes graphs, Lear lgebra ppl. 445 (4) [] J. Borwe, R. Grgesoh, class of expoetal equaltes (Preprt). [3] J.. e la Peña, I. Gutma, J. Raa, Estmatg the Estraa ex, Lear lgebra ppl. 47 (7) [4] H. Deg, S. Raekovć, I. Gutma, The Estraa ex. pplcatos of Graph Spectra, Math. Ist., Belgrae, (9) 3-4. [5] E. Estraa, The Structure of Complex Networks. Theory a pplcatos, Oxfor Uversty Press, UK,. [6] E. Estraa, J.. e la Peña, N. Hatao, Walk etropes graphs, Lear lgebra ppl. 443 (4) [7] E. Estraa, J.. Roríguez-elázquez, Subgraph cetralty complex etworks, Phys. Rev. E 7 (5) [8] E. Estraa,, N. Hatao, M. Bez, The physcs of commucablty complex etworks, Phys. Rep. 54 () [9] I. Gutma, H. Deg, S. Raekovć, The Estraa ex: a upate survey. Selecte Topcs o pplcatos of Graph Spectra, Math. Ist., Beogra, () [] B. Kostat, P. W. Mchor, The geeralze Cayley map from a algebrac group to ts Le algebra, I The orbt metho geometry a physcs, pp Brkhäuser Bosto, 3. 9