Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgra...

This paper presents an eigenclustering approach to line-segment grouping. We make three contributions. First, we show how the geometry of the lineendpoints can be used to compute a grouping field by interpolating a polar lemniscate between them. Second, we show how to adaptively threshold the grouping field to produce a line-adjacency matrix. Finally, we present a non-iterative method for locat...

We study two-path convexity in bipartite tournaments. For a bipartite tournament, we obtain both a necessary condition and a sufficient condition on the adjacency matrix for its rank to be two. We then investigate 4-cycles in bipartite tournaments of small rank. We show that every vertex in a bipartite tournament of rank two lies on a four cycle, and bipartite tournaments with a maximum number ...

Let G = G1 ∪ G2 be the sum of two simple graphs G1, G2 having a common edge or G = G1 ∪ e1 ∪ e2 ∪ G2 be the sum of two simple disjoint graphs G1, G2 connected by two edges e1 and e2 which form a cycle C4 inside G. We give a method of computing the determinant detA(G) of the adjacency matrix of G by reducing the calculation of the determinant to certain subgraphs of G1 and G2. To show the scope ...

Let μ (G) and μmin (G) be the largest and smallest eigenvalues of the adjacency matrix of a graph G. Our main results are: (i) Let G be a regular graph of order n and diameter D. If H is a proper subgraph of G, then μ (G) − μ (H) > 1 nD . (ii) If G is a connected regular nonbipartite graph of order n and diameter D, then μ (G) + μmin (G) > 1 nD .

A molecular similarity measure has been developed using molecular topological graphs and atomic partial charges. Two kinds of topological graphs were used. One is the ordinary adjacency matrix and the other is a matrix which represents the minimum path length between two atoms of the molecule. The ordinary adjacency matrix is suitable to compare the local structures of molecules such as functio...

We introduce a novel approach to description of networks/graphs. It is based on an analogue physical model which is dynamically evolved. This evolution depends on the connectivity matrix and readily brings out many qualitative features of the graph.

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle-free graphs with a given rank and characterize all such graphs achieving the maximum order.

A generalized Steinhaus graph of order n and type s is a graph with n vertices whose adjacency matrix (a i;j) satisses the relation a i;j =

In this paper, we analyze the eigenfunctions of the edge-based Laplacian on a graph and the relationship of these functions to random walks on the graph. We commence by discussing the set of eigenfunctions supported at the vertices, and demonstrate the relationship of these eigenfunctions to the classical random walk on the graph. Then, from an analysis of functions supported only on the interi...