A novel graph matrix representation: sequence of neighbourhood matrices with an application

Abstract

In the study of network optimization, finding the shortest path minimizing time/distance/cost from a source node to a destination node is one of the fundamental problems. Our focus here is to find the shortest path between any pair of nodes in a given undirected unweighted simple graph with the help of the sequence of powers of neighbourhood matrices. The authors recently introduced the concept of neighbourhood matrix as a novel representation of graphs using the neighbourhood sets of the vertices. In this article, an extension of the above work is presented by introducing a sequence of matrices, referred to as the sequence of powers of $${\mathcal {NM}}(G)$$ . It is denoted it by $${\mathcal {NM}}^{\{l\}}(G) = [\eta _{ij}^{\{l\}}], 1\le l \le k(G)$$ , where k(G) is called the iteration number, $$k(G)=\lceil \log _{2} {\hbox {diameter}}(G)\rceil$$ . As this sequence of matrices captures the distance between the nodes profoundly, we further develop the technique and present several characterizations. Based on the theoretical results, we present an algorithm to find the shortest path between any pair of nodes in a given graph. The proposed algorithm and the claims therein are formally validated through simulations on synthetic data and the real network data from Facebook. The empirical results are quite promising with our algorithm having best running time among all the existing well-known shortest path algorithms for the considered graph classes.