Walks, infinite series and spectral radius of graphs

TL;DR

This paper explores the relationship between the spectral radius of a graph and walks within its subgraphs, providing formulas for graphs containing complete multipartite spanning subgraphs, aiding spectral extremal problems.

Contribution

It introduces a new formula for the spectral radius based on infinite series of walks, specifically for graphs with complete multipartite spanning subgraphs.

Findings

Derived a formula for spectral radius using infinite series of walks.

Applicable to graphs with complete multipartite spanning subgraphs.

Contributes to spectral extremal graph theory.

Abstract

For a graph G, the spectral radius \r{ho}(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we seek the relationship between \r{ho}(G) and the walks of the subgraphs of G. Especially, if G contains a complete multi-partite graph as a spanning subgraph, we give a formula for \r{ho}(G) by using an infinite series on walks of the subgraphs of G. These results are useful for the current popular spectral extremal problem.