Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively.

With every graph (or digraph) one can associate several different matrices. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properties (the eigenvalues and eigenvectors) of these matrices provide useful information about the structure of the graph. It turns out that for regular graphs, the infor...

We give a new recursive method to compute the number of cliques and cycles of a graph. This method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. In particular, let G be a graph and let G be its complement, then given the chromatic polynomial of G, we give a r...

With every graph (or digraph) one can associate several different matrices. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properties (the eigenvalues and eigenvectors) of these matrices provide useful information about the structure of the graph. It turns out that for regular graphs, the infor...

Let G be a simple graph with adjacency matrix A(G) and π(G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Klein’s idea to compute the permanent of some matrices (Mol. Phy., 1976, Vol. 31, (3): 811−823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our mai...

In this work we attempt to answer the following question: How large a graph can we process using a vertex-centric model of computation in the main memory of a single machine? Specifically, we use a modified Pregel framework to calculate PageRank, identify connected components, and single source shortest path algorithms on two large graphs. While it is not possible to load these graphs into memo...

In this paper, we prove that for every connected graph G, there exists a split graph G' with the same independence number and the same order. Then we propose a first algorithm for finding this graph, given the degree sequence of the input graph G. Further, we propose a second algorithm for finding the independence number of G, given the adjacency matrix of G.