Let $ G $ be a connected graph with $ n $ vertices and adjacency matrix $A(G)$. The critical polynomial $d_G(x_1, \ldots, x_n) $ is a degree-$n$ multivariate polynomial defined as the determinant of the matrix $M_G(x_1, \ldots, x_n) $, where \[M_G(x_1,\ldots,x_n) = \operatorname{Diag}(x_1, \ldots, x_n) - A(G).\] For an...
Ting-Ting Wang, Lu Lu· Electronic Journal of Combin...· 0 citations
Keevash, Lenz, and Mubayi proved a spectral Erd\H{o}s--Ko--Rado theorem, showing that, for sufficiently large $n$, the complete $t$-star uniquely maximizes the adjacency-tensor spectral radius among all $t$-intersecting $k$-uniform families. In this paper, we establish a spectral Hilton--Milner--Frankl theorem for nont...
Xu-Cheng Bu, Li-Hua Feng, Lu Lu et al.· 0 citations
Let $r\ge 3$ and $k\ge 2r+1$ be fixed integers. We determine, for all sufficiently large $n$, the maximum adjacency-tensor spectral radius of an $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$. Write $s=\left\lfloor\frac{k-1}{2}\right\rfloor$. If $k=2s+1$ is odd, the unique extremal h...
Let $k>t\ge 1$ be integers and set $d=k-t$. A $k$-uniform hypergraph $\mathcal F$ is called $t$-intersecting if any two edges intersect in at least $t$ vertices, and is called $t$-critical if its minimum $t$-transversal has size $k$. Frankl proved that, for $k\ge d^4$,$|\mathcal F|\le \binom{k+d}{d},$ with equality onl...
Lu Lu, Rongrong Lu, Qifan Wang et al.· 0 citations
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