9 problems

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 $G$ 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…

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…

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