Spectral extremal problem of the $p$th power of cycles

TL;DR

This paper precisely determines the extremal graphs that maximize edges and spectral radius among large graphs avoiding the p-th power of a cycle, advancing spectral extremal graph theory.

Contribution

It provides exact extremal graphs for the maximum edges and spectral radius avoiding the p-th power of cycles, a new result in spectral extremal graph theory.

Findings

Identifies the unique extremal graph for maximum edges avoiding $C_k^p$.

Identifies the unique extremal graph for maximum spectral radius avoiding $C_k^p$.

Results hold for sufficiently large $n$.

Abstract

For a cycle $C_k$ on $k$ vertices, its $p$-th power, denoted $C_k^p$, is the graph obtained by adding edges between all pairs of vertices at distance at most $p$ in $C_k$. Let $\ex(n, F)$ and $\spex(n, F)$ denote the maximum possible number of edges and the maximum possible spectral radius, respectively, among all $n$-vertex $F$-free graphs. In this paper, we determine precisely the unique extremal graph achieving $\ex(n, C_k^p)$ and $\spex(n, C_k^p)$ for sufficiently large $n$.