COSPECTRALITY MEASURES OF GRAPHS WITH AT MOST SIX VERTICES

Cospectrality of two graphs measures the differences between the ordered spectrum of these graphs in various ways. Actually, the origin of this concept came back to Richard Brualdi s problems that are proposed in cite{braldi}: Let GnGn and G′nGn′ be two nonisomorphic simple graphs on nn vertices with spectra lambda1geqlambda2geqcdotsgeqlambdan;;;textand;;;l ambda′1geqlambda′2geqcdotsgeqlambda′n, lambda1geqlambda2geqcdotsgeqlambdan;;;textand;;;lambda1′geqlambda2′geqcdotsgeqlambdan′, respectively. Define the distance between the spectra of GnGn and G′nGn′ as lambda(Gn,G′n)=sumni=1(lambdai−lambda′i)2;:;big (textoruse;sumni=1|lambdai−lambda′i|big). lambda(Gn,Gn′)=sumi=1n(lambdai−lambdai′)2;:;big (textoruse;sumi=1n|lambdai−lambdai′|big). Define the cospectrality of GnGn by textcs(Gn)=minlambda(Gn,G′n);:;G ′n;;textnotisomorphicto;Gn.textcs(Gn)=minlambda (Gn,Gn′);:;Gn′;;textnotisomorphicto;Gn. Let textcsn=maxtextcs (Gn);:;Gn;;textagraphon;n;textvertices.textcsn=maxtextcs(Gn);:;Gn;;textagraphon;n;textvertices. Investigation of textcs(Gn)textcs(Gn) for special classes of graphs and finding a good upper bound on textcsntextcsn are two main questions in this subject. In this paper, we briefly give some important results in this direction and then we collect all cospectrality measures of graphs with at most six vertices with respect to three norms. Also, we give the shape of all graphs that are closest (with respect to cospectrality measure) to a given graph GG.