最大外平面图的最小谱半径
The minimum spectral radius of maximal outerplanar graphs
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中文总结 AI 辅助
本文证明最大外平面图的最小谱半径由之字形三角剖分F_n取得，通过三种局部操作严格减小谱半径完成证明。
中文摘要 AI 辅助
如果一个外平面图在添加任意一条边后都会失去外平面性，则称其为最大的。Lin和Ning确定了具有最大谱半径的外平面图，且该极值图是最大外平面图。我们确定了最小化者。在本文中，我们证明每个n顶点最大外平面图G满足ρ(G)≥ρ(F_n)，其中F_n是n边形的之字形三角剖分，即n个顶点上的路径的平方，且等号成立当且仅当G=F_n。证明使用了最大外平面图上的三种局部操作，每种操作都严格减小谱半径：第一种操作逆转了沿弦连接一块的方式，第二种和第三种操作分别将一块从一个顶点移动到其跨弦的双胞胎顶点，当该双胞胎不携带任何东西或携带单个耳朵时。不存在任何操作可应用的图是F_n，或其谱半径大于4，或由中心三角形和三个至少包含三个三角形的之字形叶片组成且至多有15个顶点；在最后一种情况下，它包含两个12顶点显式图之一，其谱半径超过F_15的谱半径。由于对所有n都有ρ(F_n)<4，这完成了证明。过程中使用的数值不等式由具有小条目的显式整数向量证明。
英文摘要
An outerplanar graph is \emph{maximal} if no edge can be added without losing outerplanarity. Lin and Ning determined the outerplanar graph with the largest spectral radius, and the maximizer is a maximal outerplanar graph. We determine the minimizer. In this paper, we prove that every $n$-vertex maximal outerplanar graph $G$ satisfies $ρ(G)\geρ(F_n)$, where $F_n$ is the zig-zag triangulation of the $n$-gon, that is, the square of the path on $n$ vertices, with equality if and only if $G=F_n$. The proof uses three local operations on maximal outerplanar graphs, each of which strictly decreases the spectral radius: the first reverses the way a piece is attached along a chord, and the second and third move a piece from one vertex to its twin across a chord when the twin carries nothing or a single ear, respectively. A graph at which no operation applies is $F_n$, or has spectral radius greater than $4$, or consists of a central triangle with three zig-zag blades of at least three triangles each and has at most $15$ vertices; in the last case it contains one of two explicit graphs on $12$ vertices whose spectral radius exceeds that of $F_{15}$. Since $ρ(F_n)<4$ for all $n$, this completes the proof. The numerical inequalities used along the way are certified by explicit integer vectors with small entries.
发表机构
- The State University of New York, Korea（纽约州立大学韩国分校）
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