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

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 $\ddot{I}(G)\le \ddot{I}{}^{\prime }(m)$, where $\ddot{I}{}^{\prime }(m)$ 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.

39 pages, includes a computational appendix