The Sharp Even-Size Spectral Threshold for $H(4,3)$-Free Graphs

TL;DR

This paper establishes the exact spectral threshold for $H(4,3)$-free graphs with even size, identifying the extremal graphs and refining the Perron-neighborhood method for the proof.

Contribution

It determines the sharp spectral threshold for $H(4,3)$-free graphs and introduces a refined proof technique using local interface independence.

Findings

The threshold is given by the largest root of a specific quartic polynomial.

Equality characterizes the extremal graph $S^-_{(m+4)/2,2}$.

Explicit obstruction graphs show the threshold is sharp at size 18.

Abstract

We determine the sharp even-size threshold for the fixed-size spectral extremal problem forbidding $H(4,3)$, the graph obtained by identifying one vertex of a $4$-cycle with one vertex of a triangle. Specifically, if $G$ is an $H(4,3)$-free graph of even size $m \ge 18$ with no isolated vertices, then $\rho(G) \le \rho'(m)$, where $\rho'(m)$ is the largest real root of $x^4 - m x^2 - (m-2)x + m/2 - 1 = 0$. Equality holds if and only if $G \cong S^-_{(m+4)/2,2}$. The value $18$ is best possible: explicit $H(4,3)$-free obstruction graphs exceed the comparison value for $m = 10,12,14,16$. The proof refines the Perron-neighborhood method by proving a local interface independence principle in the $K_4$-core branch, reducing the remaining threshold cases to finite endpoint comparisons.