非均衡符号多部图的最大指标与谱半径
The maximum index and spectral radius of unbalanced signed multipartite graphs
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中文总结 AI 辅助
本文研究非均衡符号多部图的指标与谱半径极值问题，确定了固定t≥2及部集大小的非均衡符号t部图中，切换同构意义下分别最大化指标与谱半径的情况，同时确定了对应最小特征值最小的情况。
中文摘要 AI 辅助
设Γ=(G,σ)为符号图，其中G为底图，顶点集为V(G)，边集为E(G)，σ:E(G)→{-1,1}为符号函数。对U⊂V(G)，将U与V(G)\backslash U之间所有边的符号翻转的操作称为切换。具有相同底图的两个符号图，若可通过对某个子集切换从一个得到另一个，则称它们切换等价；若其中一个与另一个的某个切换等价符号图同构，则称它们切换同构。符号环若包含奇数条负边则称为负环；若没有负环则符号图是均衡的，否则为非均衡的。Γ的邻接矩阵A(Γ)由G的标准(0,1)邻接矩阵构造，将对应负边的所有1的符号翻转得到。Γ的指标是A(Γ)的最大特征值，谱半径是A(Γ)的特征值的最大绝对值，最小特征值是A(Γ)的最小特征值。我们研究非均衡符号多部图中指标与谱半径的极值问题，更准确地说，确定固定t≥2且部集大小（阶）的非均衡符号t部图，在切换同构意义下分别最大化指标与谱半径的情况；为确定固定部集大小（阶）的非均衡符号多部图中谱半径最大的情况，我们还确定了其中最小特征值最小的情况。
英文摘要
Let $Γ=(G,σ)$ be a signed graph, where $G$ is the underlying graph with vertex set $V(G)$ and edge set $E(G)$ such that $σ: E(G)\to \{-1,1\}$ is the sign function. For $U\subset V(G)$, the operation that changes the sign of all edges between $U$ and $V(G)\setminus U$ is called switching. Two signed graphs with the same underlying graph are switching equivalent if one is obtainable from the other one by switching a subset. Two signed graphs are switching isomorphic if one is isomorphic to a switching equivalent signed graph of the other one. A signed cycle is called negative if it contains an odd number of negative edges. A signed graph is balanced if none of its cycles is negative; otherwise it is unbalanced. The adjacency matrix $A(Γ)$ of $Γ$ is obtained from the standard $(0,1)$-adjacency matrix of $G$ by reversing the sign of all $1$s which correspond to negative edges. The index of $Γ$ is the largest eigenvalue of $A(Γ)$ and the spectral radius of $Γ$ is the largest absolute value of the eigenvalue of $A(Γ)$. The least eigenvalue of $Γ$ is the least eigenvalue of $A(Γ)$. We study the extremal problems of the index and the spectral radius among unbalanced signed multipartite graphs. More precisely, we determine the unbalanced signed $t$-partite graphs with fixed $t\ge 2$ and partite sizes (order, respectively) that maximizes the index and the spectral radius respectively, up to switching isomorphism. To determine the unbalanced signed multipartite graphs with fixed partite sizes (order, respectively) with maximum spectral radius, we also determine those with minimum least eigenvalue.
