The Sharp Even-Size Spectral Threshold for $H(4,3)$-Free Graphs
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913119576522752 |
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| author | Sarkar, Shreyhaan |
| author_facet | Sarkar, Shreyhaan |
| contents | 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 $ρ(G) \le ρ'(m)$, where $ρ'(m)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19854 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Sharp Even-Size Spectral Threshold for $H(4,3)$-Free Graphs Sarkar, Shreyhaan Combinatorics 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 $ρ(G) \le ρ'(m)$, where $ρ'(m)$ 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. |
| title | The Sharp Even-Size Spectral Threshold for $H(4,3)$-Free Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.19854 |