论文标题
互补的消失图
Complementary Vanishing Graphs
论文作者
论文摘要
Given a graph $G$ with vertices $\{v_1,\ldots,v_n\}$, we define $\mathcal{S}(G)$ to be the set of symmetric matrices $A=[a_{i,j}]$ such that for $i\ne j$ we have $a_{i,j}\ne 0$ if and only if $v_iv_j\in e（g）$。由图形补体猜想的动机，我们说，如果存在矩阵$ a \ in \ mathcal {s}（g）$和$ b \ in \ Mathcal {s}（\ overline {g}）$，则图形$ g $是互补的消失。我们为图形是或不是互补消失的何时提供组合条件，并且根据某些最小互补的消失图，哪些图是互补消失的。除此之外，我们确定最多$ 8 $顶点的哪些图是互补的。
Given a graph $G$ with vertices $\{v_1,\ldots,v_n\}$, we define $\mathcal{S}(G)$ to be the set of symmetric matrices $A=[a_{i,j}]$ such that for $i\ne j$ we have $a_{i,j}\ne 0$ if and only if $v_iv_j\in E(G)$. Motivated by the Graph Complement Conjecture, we say that a graph $G$ is complementary vanishing if there exist matrices $A \in \mathcal{S}(G)$ and $B \in \mathcal{S}(\overline{G})$ such that $AB=O$. We provide combinatorial conditions for when a graph is or is not complementary vanishing, and we characterize which graphs are complementary vanishing in terms of certain minimal complementary vanishing graphs. In addition to this, we determine which graphs on at most $8$ vertices are complementary vanishing.
