COSPECTRALITY MEASURES OF GRAPHS WITH AT MOST SIX VERTICES

Abstract

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 $G_n$ and $G'_n$ be two nonisomorphic simple graphs on $n$ vertices with spectra<br />$$lambda_1 geq lambda_2 geq cdots geq lambda_n ;;;text{and};;; lambda'_1 geq lambda'_2 geq cdots geq lambda'_n,$$ respectively. Define the distance between the spectra of $G_n$ and $G'_n$ as<br />$$lambda(G_n,G'_n) =sum_{i=1}^n (lambda_i-lambda'_i)^2 ;;; big(text{or use}; sum_{i=1}^n|lambda_i-lambda'_i|big).$$<br />Define the cospectrality of $G_n$ by<br />$text{cs}(G_n) = min{lambda(G_n,G'_n) ;:; G'_n ;;text{not isomorphic to} ; G_n}.$<br />Let $text{cs}_n = max{text{cs}(G_n) ;:; G_n ;;text{a graph on}; n ;text{vertices}}.$<br />Investigation of $text{cs}(G_n)$ for special classes of graphs and finding a good upper bound on $text{cs}_n$ are two main questions in this<br />subject.<br />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 $G$.