Let G be a simple graph of order n with eigenvalues λ1, λ2, · · · , λn and normalized Laplacian eigenvalues μ1,μ2, · · · ,μn. The Estrada index and normalized Laplacian Estrada index are defined as EE(G) = ∑n k=1 e λk and LEE(G) = ∑n k=1 e μk−1, respectively. We establish upper and lower bounds to EE and LEE for edge-independent random graphs, containing the classical Erdös-Rényi graphs as spec...

let g be a (p, q) graph. let f : v (g) → {1, 2, . . . , k} be a map. for each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of g if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

The distance matrix of a simple connected graph G is D(G) = (dij), where dij is the distance between ith and jth vertices of G. The multiset of all eigenvalues of D(G) is known as the distance spectrum of G. Lin et al.(On the distance spectrum of graphs. Linear Algebra Appl., 439:1662-1669, 2013) asked for existence of graphs other than strongly regular graphs and some complete k-partite graphs...

In this paper we study the complementarity spectrum of digraphs, with special attention to problem digraph characterization through spectrum. That is, whether two non-isomorphic digraphs same number vertices can have eigenvalues. The eigenvalues matrices, also called Pareto eigenvalues, has led (undirected) graphs and, in particular, undirected these is an open problem. We characterize one and ...

The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many of the commonly studied spectra of graphs. We show that for a large class of graphs these eigenvalues can be computed explicitly. We also present the applicat...

One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplacian matrix. Determining this number is of theoretical interest and also of practical impact. Graphs with small spectra exhibit many symmetry properties and are well suited as interconnection topologies. Especially load balancing can be done on such interconnection topologies in a small n...

Spectral graph theory looks at the interplay between the structure of a graph and the eigenvalues of a matrix associated with the graph. Many interesting graphs have rich structure which can help in determining the eigenvalues associated with some particular matrix of a graph. This survey looks at some common techniques in working with and determining the eigenvalues associated with the normali...

One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplace matrix. Determining this number is of theoretical interest as well as of practical impact. Sparse graphs with small spectra exhibit excellent structural properties and can act as interconnection topologies. In this paper, for any n we present graphs, for which the product of their vert...