Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~$G=K_{(3,\,2^\beta)}$ are $e$-positive. We resolve this question and, together with their classification, characterize all $e$-positive complete multipartite graphs. Our main result is an explicit, manifestly nonnegative $e$-expansion of~$X_G$ whose coefficients are expressed in terms of the restricted-injection numbers. Our main idea is to derive a marker-variable coefficient-extraction formula for the $e$-coefficients of arbitrary complete multipartite graphs from the elementary--monomial Cauchy identity. For the particular graph~$G$, this formula reduces the proof to three coefficient families, which we evaluate using Dickson polynomials and recurrences for these numbers.
For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, e...
Motivated by Brouwer's conjecture, Ashraf, Omidi and Tayfeh-Rezaie proposed the following Brouwer-type conjecture that for every graph $G$ on $n$ vertices with $m$ edges, the sum $S_k^+(G)$ of its $k$ largest signless Laplacian eigenvalues satisfies $S_k^+(G)\le m+\binom{k+1}{2}$ for $k=1, \ldots, n$. In this paper, we...
We study the hard-core model on recursively defined sequences $(G_n)_{n\geq0}$ of graphs with a fixed number $k\geq 1$ of labeled vertices in each graph. The next graph in the sequence is constructed by taking a fixed number $m\geq 2$ of copies of the previous graph, connecting these copies by identifying some labeled...
For $n\ge6$, $m=\binom n2$, let the complete-graph incidence family on $K_n$ have the vertices of $K_n$ as columns, its edges as rows, and the incidence graph as one-edge graph. The universal cell bound of L\"ofberg and Qi gives $z_2(m,n)\le Z(n):=\lfloor n(n-1)(n+2)/4\rfloor$. We implement the nested one-factorization...
Yan-Nan Chen, J. Löfberg, Li-Qun Qi· 2 citations· ⚡1
For a homogeneous $A$-hypergeometric system, let $K_v$ be the intersection of the negative support family for a fake exponent $v$, and let $C_w$ be the set of indices of variables that do not occur in any minimal monomial generator of $\operatorname{in}_w(I_A)$. We prove that $K_v=\operatorname{nsupp}(v)\cap C_w$ holds...
Let $\mathcal{B}$ be the class consisting of the six-vertex bipartite graphs that possess a perfect matching and their complements. It is proved that every $\mathcal{B}$-free graph $G$ satisfies $\alpha(G)+\omega(G)\ge |V(G)|-1$. This establishes Conjecture 3.1 of Litjens, Polak and Sivaraman (B-Free Graphs Conjecture)...
Domenico Frijio· 0 citations
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