The spectrum of a matrix M is the multiset that contains all the eigenvalues of M. If M is a matrix obtained from a graph G, then the spectrum of M is also called the graph spectrum of G. If two graphs has the same spectrum, then they are cospectral (or isospectral) graphs. In this paper, we compare four spectra of matrices to examine their accuracy in protein structural comparison. These four ...

Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention researchers recent years. It generally very hard and challenging to show given determined its spectrum. In Wang (2017), the author gave simple arithmetic condition for family being generalized spectra. However, method applies only so called controllable ; it fails when a...

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the ...

This is a new contribution to the question: Can we see from the spectrum of a graph whether it is distance-regular? By generalizing some results of Van Dam and Haemers, among others, we give some new spectral and quasi-spectral characterizations of distance-regularity. In this area of research, typical results concluding that a graph is distance regular require that G is cospectral with a dista...

A dumbbell graph, denoted by Da,b,c, is a bicyclic graph consisting of two vertexdisjoint cycles Ca, Cb and a path Pc+3 (c > −1) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of Da,b,0 (without cycles C4) with gcd(a, b) > 3, and we complete the research started in [J.F. Wang et al., A ...

The energy, E(G), of a simple graph G is defined to be the sum of the absolute values of the eigen values of G. If G is a k-regular graph on n vertices,then E(G) k+√k(n− 1)(n− k)= B2 and this bound is sharp. It is shown that for each > 0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k < n− 1 and B2 < . Two graphs with the same number of vertic...

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The integral circulant graph ICGn(D) has the vertex set Zn = {0, 1, 2, . . . , n− 1} and vertices a and b are adjacent if gcd(a− b, n) ∈ D, where D ⊆ {d : d | n...

The problem of finding a perfect matching in an arbitrary simple graph is well known and popular theory. It used various fields, such as chemistry, combinatorics, game theory etc. M G set pairwise nonadjacent edges, ie, those that do not have common vertices. Matching called if it covers all vertices the graph, ie each incidental to exactly one edges. By Koenig's theorem, regular bipartite grap...