发表机构
Mathematical Institute in Opava, Silesian University in Opava（奥帕瓦数学研究所，奥帕瓦西里西亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究强连通有向图类的最小谱半径，通过生成函数和耳朵分解给出结构分类，证明最小谱半径由特定多项式决定，并确定极值图及上下界。
AI 中文摘要
我们研究了具有 $m$ 个顶点和 $m+2$ 条边的强连通有向图类 $\mathcal{SC}_{m+2}(m)$ 中的最小邻接谱半径。利用有向路径的生成函数，我们将相关有向图与拓扑多项式联系起来，这些多项式的最小正根决定了相应的谱半径。基于耳朵分解，我们通过证明该类中的每个有向图都可以通过附加单个耳朵从蝴蝶有向图得到，从而获得了 $\mathcal{SC}_{m+2}(m)$ 的完整结构分类。这将极值问题简化为在可实现性条件下对有限多个多项式族的优化和比较。我们证明了最小谱半径由多项式 $P_{\min}(z)=1-2z^{m-1}-z^m$ 决定。若 $R_m\in(0,1)$ 表示其唯一根，则 $\min_{G\in\mathcal{SC}_{m+2}(m)}\rho(G)=R_m^{-1}$。对于 $m\geq4$，最小值在同构意义下唯一地由交叉弦环 $\mathcal{C}_m^\times$ 取得。对于 $m=3$，恰好有两个非同构的极小化图，它们的谱半径均为 $(1+\sqrt5)/2$。最后，我们建立了界 $2^{1/(m-1)}<\rho\left(\mathcal{C}_m^\times\right)<3^{1/(m-1)}$。
英文摘要
We study the minimum adjacency spectral radius in the class $\mathcal{SC}_{m+2}(m)$ of strongly connected digraphs with $m$ vertices and $m+2$ edges. Using generating functions for directed paths, we associate with the relevant digraphs topological polynomials whose smallest positive roots determine the corresponding spectral radii. Based on an ear decomposition, we obtain a complete structural classification of $\mathcal{SC}_{m+2}(m)$ by showing that every digraph in this class can be obtained from a butterfly digraph by attaching a single ear. This reduces the extremal problem to the optimization and comparison of finitely many polynomial families subject to their realizability conditions. We prove that the minimum spectral radius is determined by the polynomial $P_{\min}(z)=1-2z^{m-1}-z^m$. If $R_m\in(0,1)$ denotes its unique root, then $\min_{G\in\mathcal{SC}_{m+2}(m)}ρ(G)=R_m^{-1}$. For $m\geq4$, the minimum is attained, up to isomorphism, uniquely by the cross-chorded cycle $\mathcal{C}_m^\times$. For $m=3$, there are exactly two non-isomorphic minimizers, both with spectral radius $(1+\sqrt5)/2$. Finally, we establish the bounds $2^{1/(m-1)}<ρ\left(\mathcal{C}_m^\times\right)<3^{1/(m-1)}$.
Comments50 pages, 18 figures
