With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices pro...

In general, classifying graphs with labelled nodes (also known as labelled graphs) is a more difficult task than classifying graphs with unlabelled nodes. In this work, we decompose the labelled graphs into unlabelled subgraphs with respect to the labels, and describe these decomposed subgraphs with the travelling matrices. By utilizing the travelling matrices to calculate the dissimilarity for...

We obtain formulae for group inverses of matrices that are associated with a new class digraphs obtained from stars. This contains both bipartite and non-bipartite graphs. Expressions the inverse corresponding to double star adjacency matrix certain undirected multi-star graphs also proven. A blockwise representation or Dutch windmill graph is presented.

In this note we show how to construct two distinct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices by “unfolding” a base bipartite graph in two different ways.

We compute the elementary divisors of the adjacency and Laplacian matrices of the Grassmann graph on 2-dimensional subspaces in a finite vector space. We also compute the corresponding invariants of the complementary graphs.

This paper deals with adjacency matrices of signed cycle graphs and chemical descriptors based on them. The eigenvalues and eigenvectors of the matrices are calculated and their efficacy in classifying different signed cycles is determined. The efficacy of some numerical indices is also examined. Mathematics Subject Classification 2010: Primary 05C22; Secondary 05C50, 05C90

We count invertible Schrödinger operators (perturbations by diagonal matrices of the adjacency matrix) over finite fields for trees, cycles and complete graphs. This is achieved for trees through the definition and use of local invariants (algebraic constructions of perhaps independent interest). Cycles and complete graphs are treated by ad hoc methods.

Computational techniques are described for the automorphism groups of edge-weighted graphs. Fortran codes based on the manipulation of weighted adjacency matrices are used to compute the automorphism groups of several edge-weighted graphs. The code developed here took 37l/2 min of CPU time to generate 1 036 800 permutations in the automorphism group of an edge-weighted graph.

Motivated from an example of ridge graphs relating to metric polytopes, a class of connected regular graphs such that the squares of their adjacency matrices are in certain symmetric Bose-Mesner algebras of dimension 3 is considered in this paper as a generalization of strongly regular graphs. In addition to analysis of this prototype example defined over ðMetP5Þ , some general properties of th...