Spectral extremal problem for the odd prism

TL;DR

This paper determines the maximum spectral radius of large graphs that do not contain an odd prism, identifying the unique extremal graph for such spectral extremal problems.

Contribution

It establishes the spectral Turán number for the odd prism in large graphs and characterizes the unique extremal graph.

Findings

Exact value of spectral Turán number for odd prism

Identification of the unique extremal graph for large n

Extension of spectral extremal theory to Cartesian products

Abstract

The spectral Tur\'an number $\spex(n, F)$ denotes the maximum spectral radius $\rho(G)$ of an $F$-free graph $G$ of order $n$. This paper determines $\spex\left(n, C_{2k+1}^{\square}\right)$ for all sufficiently large $n$, establishing the unique extremal graph. Here, $C_{2k+1}^{\square}$ is the odd prism -- the Cartesian product $C_{2k+1} \square K_2$ -- where the Cartesian product $G \square F$ has vertex set $V(G) \times V(F)$, and edges between $(u_1,v_1)$ and $(u_2,v_2)$ if either $u_1 = u_2$ and $v_1v_2 \in E(F)$, or ($v_1 = v_2$ and $u_1u_2 \in E(G)$).