An Unique and Novel Graph Matrix for Efficient Extraction of Structural Information of Networks

TL;DR

This paper introduces a new neighbourhood matrix for undirected graphs that captures local and extended neighborhood information, enabling more efficient analysis and reconstruction of network structures.

Contribution

The paper proposes a novel neighbourhood matrix that encodes local and second-level neighborhood information, offering improved efficiency and new insights for graph analysis.

Findings

The neighbourhood matrix exhibits a bijection with the product of adjacency and Laplacian matrices.

It can be used to reconstruct the original graph from the matrix.

The matrix enables solving graph problems with reduced time complexity.

Abstract

In this article, we propose a new type of square matrix associated with an undirected graph by trading off the naturally imbedded symmetry in them. The proposed matrix is defined using the neighbourhood sets of the vertices. It is called as neighbourhood matrix and it is denoted by $ \mathcal{NM}(G)$ as this proposed matrix also exhibits a bijection between the product of the two graph matrices, namely the adjacency matrix and the graph Laplacian. This matrix can also be obtained by looking at every vertex and the subgraph with vertices from the first two levels in the level decomposition from that vertex. The two levels in the level decomposition of the graph give us more information about the neighbour of a vertex along with the neighbour of neighbour of a vertex. This insight is required and is found useful in studying the impact of broadcasting on social networks, in particular, and complex networks, in general. We establish several interesting properties of the $ \mathcal{NM}(G) $. In addition, we also show how to reconstruct a graph $G$, given a $ \mathcal{NM} (G)$. The proposed matrix is also found to solve many graph theoretic problems using less time complexity in comparison to the existing algorithms.