具有相同组合不变量和不同Lovász数的连通非正则同谱图
Connected irregular cospectral graphs with identical combinatorial invariants and distinct Lovász numbers
- Faculty of Mathematics, Technion–Israel Institute of Technology（以色列理工学院数学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对每个n≥10构造连通非正则非同构同谱图对，它们共享四种谱和四个组合不变量但Lovász数不同，最小阶数为10。
AI中文摘要：
对于每个整数$n \geq 11$，我们构造一对连通、非正则、非同构的图，每个图有$n$个顶点，它们在邻接矩阵、Laplacian矩阵、无符号Laplacian矩阵和归一化Laplacian矩阵下是同谱的，并且具有相同的独立数、团数、色数和补图的色数，但具有不同的Lovász $\vartheta$-数，且这些数与$n$无关。对于$n=10$，我们首先展示一对满足类似条件的正则图对。我们的构造将这一固定的十顶点正则图对的每个成员与一个完全图进行普通联图，从而保持关于所有四个矩阵的同时同谱性以及各自的Lovász数。我们推导了这些Lovász数的精确解析表达式，并证明它们不同，而不依赖数值近似。穷举搜索表明，在少于十个顶点的连通非正则非同构图对中，不存在同时关于四个矩阵同谱且具有相同四个列出的组合不变量的图对。此外，搜索产生了一个十顶点上的连通非正则非同构图对，满足所有这些条件且具有不同的Lovász $\vartheta$-数。因此，存在这样的连通非正则图对的最小阶数恰好是10，并且对于每个$n \geq 10$都存在这样的图对。这加强了先前关于所有偶数阶$n \geq 14$的存在性结果（Sason, 2024）。除了在界定Shannon容量和其他组合图不变量方面的作用外，Lovász数因此为一系列无法通过四个谱或四个列出的不变量区分的图对提供了可有效计算的非同构证书。
英文摘要：
For every integer $n\geq 11$, we construct pairs of connected, irregular, nonisomorphic graphs on $n$ vertices that are cospectral for the adjacency, Laplacian, signless Laplacian, normalized Laplacian, and Seidel matrices, have equal independence, clique, chromatic, complement chromatic, and maximum-cut numbers, and have distinct Lovász $\vartheta$-numbers. Each pair is formed by joining $K_{n-10}$ to fixed cospectral, nonisomorphic, regular graphs on ten vertices due to van Dam and Haemers. We prove that the joins retain equality of the five spectra and listed integer-valued invariants, while preserving the respective Lovász numbers. We derive exact formulas for these numbers and prove them distinct. We also determine the cardinality-constrained maximum-cut profiles of the base graphs and their complements. They give exact formulas and prove equality of the maximum-cut numbers within each pair, both for the joins of the base graphs with $K_{n-10}$ and for those of their complements with $K_{n-10}$. For $n=10$, we first give a regular pair with all the stated properties except irregularity, then a connected, irregular, nonisomorphic pair sharing all five spectra and listed integer-valued invariants but having distinct Lovász numbers. An exhaustive SageMath computation shows that no connected, irregular, nonisomorphic pair on at most nine vertices shares all five spectra and listed integer-valued invariants. Thus, ten is the smallest possible order, and such pairs exist for every $n\geq 10$. It extends and strengthens a result for even $n\geq 14$ (Sason, '24), which did not address Seidel matrices, complement chromatic numbers, maximum-cut numbers of the graphs, or those of the corresponding joins formed from their complements. Thus, the Lovász number is a computable certificate of nonisomorphism even when all five spectra and listed integer-valued invariants coincide.
