It is shown that only a fraction of 2−Ω(n) of the graphs on n vertices have an integral spectrum. Although there are several explicit constructions of such graphs, no upper bound for their number has been known. Graphs of this type play an important role in quantum networks supporting the so-called perfect state transfer. AMS Subject Classification: 05C50, 05C80

The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. S...

The asymptotic spectrum of graphs, introduced by Zuiddam (Combinatorica, 2019), is the space graph parameters that are additive under disjoint union, multiplicative strong product, normalized and monotone homomorphisms between complements. He used it to obtain a dual characterization Shannon capacity graphs as minimum evaluation function over noted several known upper bounds, including Lovász n...

A disjoint union of complete graphs is in general not determined by its Laplacian spectrum. It is shown in this paper that if one only considers the family of graphs without isolated vertex, then a disjoint union of complete graphs is determined by its Laplacian spectrum within this family. Moreover, it is shown that the disjoint union of two complete graphs with a and b vertices, a b > 5 3 and...

We study the spectrum of join several circulant matrices. apply our results to compute explicitly certain graphs obtained by joining graphs.

Abstract Graphs can be associated with a matrix according to some rule and we find the spectrum of graph respect that matrix. Two graphs are cospectral if they have same spectrum. Constructions help us establish patterns about structural information not preserved by We generalize construction for previously given distance Laplacian larger family graphs. In addition, show appropriate assumptions...

Consider two graphs G and H. Let H[G] be the lexicographic product of H and G, where H is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H[G] and H whenG andH are regular and the Laplacian spectrum ofH[G] andH for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic p...