16 problems

Let $G$ be an $r$-regular bipartite graph with $v=2n$ vertices, let $m_i$ denote the number of $i$-matchings, and define … For the finite-difference operator defined by…

Csikvári's conjecture. The extremal graph maximizing the largest matching root is as star-like' as possible: it has as many vertices of degree $n-1$ as possible, and one more verte…

Csikvári's conjecture. The extremal graph is as star-like' as possible: it has as many vertices of degree $n-1$ as possible, and one more vertex of the clique part of the threshold…

Let $G$ be a connected graph, let $F\subseteq E(G^c)$ be nonempty, and set $H=G+F$. Let $\lambda_1(G),\dots,\lambda_n(G)$ denote the roots of the Laplacian matching polynomial of…

Let $G$ be a connected graph and let $e$ be a non-edge of $G$. Write the nonnegative real roots of the Laplacian matching polynomial $4\mathscr{LM}_G(x)\4$ in non-increasing order…

Let $\mathcal{H}$ be an $r$-tree with $r\geq 3$, and let $H$ be a subgraph induced by some vertex subset. Let $\varphi_H$ denote the matching polynomial of $H$, and let…

Let $(T,\mathbf{w})$ be a weighted $k$-tree, and let $T'$ be a subtree of $T$. Write $\mu(T',\mathbf{w},x)$ for the weighted matching polynomial of $T'$ and let…

Let $G$ be an $r$-regular bipartite graph with $v=2n$ vertices, let $m_i$ denote the number of $i$-matchings, and define … Let the finite-difference operator satisfy … The graph sa…

Let $G$ be a $d$-regular graph and let $Z_G^m(\lambda)$ be its matching polynomial. For even $d$, the preceding clique-minimization claim proposes that $K_{d+1}$ is the relevant mi…

Let $G$ be a $d$-regular graph, let $Z_G^m(\lambda)=\sum_{k\ge0}m_k(G)\lambda^k$ be its matching polynomial, and let $K_{d+1}$ be the complete graph on $d+1$ vertices. For even $d$…

Consider $r$-regular bipartite graphs with $v=2n$ vertices. Let $m_i$ be the number of $i$-matchings, and let $\overline{m}_i$ be the number of $i$-matchings in the complete graph…

Let $C_n$ be the cycle graph on $n$ vertices, let $P_m$ be the path graph on $m$ vertices, and let $d4d0_{G,d}(x)$ denote the $d$-matching polynomial of a graph $G$, defined as the…

Let $K_{(n)}$ be the complete $k$-partite graph with $N=n_1+dots+n_k$ vertices, whose $k$ color classes have sizes $n_1,dots,n_k$. For odd $r$, let $M_r(K_{(n)})$ denote its matchi…

Let $C$ be a circuit contained in a graph $G$. The matching polynomial is denoted by $\alpha(G,x)$, and the two circuit characteristic, or beta-polynomials, considered here are … a…

Looped-graph conjecture. Every such $G$ has a one-sided Ramanujan $r$-covering; $\mathcal{M}_{d,G}$ is real-rooted for every $d$; and $G$ has a one-sided Ramanujan $(\Gamma,\pi)$-c…

Cavers et al.'s conjecture. If $G$ is an odd-cycle graph of order $n$, then