Let G be a finite group with |G|?4 and S subset of |S|=d such that the Cayley sum graph C ? (G,S) is undirected connected. We show non-trivial spectrum normalised adjacency operator controlled by its Cheeger constant degree. establish an explicit lower bound for these graphs, namely, eigenvalues lies in interval - 1 + h (G) 4 ? , 2 2d where denotes vertex d-regular ?=2 9 d 8 . Further, we impro...

If G is a graph, its Laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix. The main thrust of the present article is to prove several Laplacian eigenvector “principles” which in certain cases can be used to deduce the effect on the spectrum of contracting, adding or deleting edges and/or of coalescing vertices. One application is the construction of ...

Two graphs G and H are called R-cospectral if A(G)+yJ and A(H)+yJ (where A(G), A(H) are the adjacency matrices of G and H, respectively, J is the all-one matrix) have the same spectrum for all y ∈ R. In this note, we give a necessary condition for having R-cospectral graphs. Further, we provide a sufficient condition ensuring only irrational orthogonal similarity between certain cospectral grap...

Random matrix theory is finding an increasing number of applications in the context of information theory and communication systems, especially in studying the properties of complex networks. Such properties include short-term and long-term correlation. We study the spectral fluctuations of the adjacency of networks using random-matrix theory. We consider the influence of the spectral unfolding...

Visualizing network data is applicable in domains such as biology, engineering, and social sciences. We report the results of a study comparing the effectiveness of the two primary techniques for showing network data: node-link diagrams and adjacency matrices. Specifically, an evaluation with a large number of online participants revealed statistically significant differences between the two vi...

We obtain a sequence k,(G) Q k,(G) < *. . < k,(G) of lower bounds for the clique number (size of the largest clique) of a graph G of n vertices. The bounds involve the spectrum of the adjacency matrix of G. The bound k,(G) is explicit and improves earlier known theorems. The bound k,(G) is also explicit, and is shown to improve on the bound from Brooks’ theorem even for regular graphs. The boun...

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...