Linear Algebra and Number of Spanning Trees

TL;DR

This paper introduces a new algebraic method linking linear algebra and graph theory to compute the number of spanning trees, demonstrated on specific networks, and discusses potential improvements in determinant calculation.

Contribution

It presents a novel algebraic approach for deriving formulas for the complexity of certain networks using linear algebra techniques.

Findings

Derived explicit formulas for the number of spanning trees in specific networks

Compared spanning trees entropy between different network structures

Proposed an open problem to improve determinant calculation methods

Abstract

A network-theoretic approach for determining the complexity of a graph is proposed. This approach is based on the relationship between the linear algebra (theory of determinants) and the graph theory. In this paper we contribute a new algebraic method to derive simple formulas of the complexity of some new networks using linear algebra. We apply this method to derive the explicit formulas for the friendship network and the subdivision of friendship graph . We also calculate their spanning trees entropy and compare it between them. Finally, we introduce an open problem "Any improvement for calculating of the determinant in linear algebra, we can investigate this improvement as a new method to determine the number of spanning tree for a given graph.