Some lemmas on spectral radius of graphs: including an application

TL;DR

This paper presents new lemmas on the spectral radius of graphs containing a spanning complete bipartite subgraph and applies these results to identify extremal planar graphs with maximum spectral radius avoiding certain cycle lengths.

Contribution

The paper introduces three new lemmas on the spectral radius for graphs with a spanning complete bipartite graph and characterizes extremal planar graphs with maximum spectral radius under cycle length constraints.

Findings

Established three lemmas relating spectral radius and spanning bipartite subgraphs.

Characterized the unique extremal planar graph with maximum spectral radius avoiding cycles of length .

Applied spectral methods to solve extremal graph problems.

Abstract

For a graph $G$, the spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we give three lammas on $\rho(G)$ when $G$ contains a spanning complete bipartite graph. Using these lemmas and typical spectral method, we characterized the unique extremal graph with the maximum spectral radius among all planar graphs of large order $n$ without a cycle of length $\ell$, where $5\leq \ell\leq n$.