A complete subgraph of a given graph is called a clique. A clique Polynomial of a graph is a generating function of the number of cliques in $G$. A real root of the clique polynomial of a graph $G$ is called a \emph{clique root} of $G$. \\ Hajiabolhassan and Mehrabadi showed that the clique polynomial of any simple graph has a clique root in $[-1,0)$. As a generalization of their result, the author of this paper showed that the class of $K_{4}$-free connected chordal graphs has also only clique roots. \\ A given graph $G$ is called flat if each edge of $G$ belongs to at most two triangles of $G$. In answering the author's open question about the class of \emph{non-chordal} graphs with the same property of having only c;ique roots, we extend the aforementioned result to the class of $K_{4}$-free flat graphs. In particular, we prove that the class of $K_{4}$-free flat graphs without isolated edges has $r=-1$ as one of its clique roots. We finally present some interesting open questions and conjectures regarding clique roots of graphs.
翻译:给定图形的完整子图称为 cliques 。 一个图形的 Coltic 集合点是一个以 G$ 生成 cliques 数函数的函数。 一个图形的 clique $G$ 的圆形多元值的真正根根叫做 $G$ 的 emph{ clique root} $G$ 。\ Hajiabolhassan 和 Mehrabadi 显示, 任何简单图形的圆形多元值在 $[1,0] 中有一个圆形根。 作为结果的概括, 本文作者显示, 无 $ $ 4} 的 根连接的 chaordal 图形也只具有结晶根根 。\\ a 给定点$ g$ $ g$ 如果$ 的每个边缘都属于$G$ 最多两个三角, 。 在回答作者关于 \ emph{ noncordal} 图形的开放属性时, 我们将上述结果推广到 $ $4} 和 平端平面的平面的平面图的类。