Spectral radius and the $2$-power of Hamilton cycles

TL;DR

This paper identifies the unique graph with the maximum spectral radius among all graphs of a given size that do not contain the 2-power of a Hamilton cycle, advancing spectral graph theory understanding.

Contribution

It determines the unique extremal graph with maximum spectral radius avoiding the 2-power of a Hamilton cycle for any order n.

Findings

Identifies the extremal graph with maximum spectral radius

Characterizes the structure of graphs excluding the 2-power of Hamilton cycles

Advances spectral extremal graph theory

Abstract

Let $G$ be a graph of order $n$ and spectral radius be the largest eigenvalue of its adjacency matrix, denoted by $\mu(G)$. In this paper, we determine the unique graph with maximum spectral radius among all graphs of order $n$ without containing the $2$-power of a Hamilton cycle.