由广义谱确定的几乎可控图的构造
Constructions of almost controllable graphs determined by their generalized spectra
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中文总结 AI 辅助
本文基于Lin等人的DGS准则，通过研究改进的行走矩阵在图操作下的演化，构造了无限族几乎可控DGS图，扩展了Liu等人的相关构造。
中文摘要 AI 辅助
在谱图理论中，识别和构造由广义谱确定的图（DGS）是一项重要且具有挑战性的问题。Lin等人（2026）利用改进的行走矩阵，提出了几乎可控图为DGS的简单准则。本文研究改进的行走矩阵在不交并与单顶点连接操作下的演化，建立了所得图的改进行走矩阵行列式的精确代数恒等式。基于该恒等式和Lin等人的DGS准则，本文构造了无限族几乎可控DGS图，扩展了Liu等人（2019）仅针对可控图的先前构造。
英文摘要
Identifying and constructing graphs that are determined by their generalized spectrum (DGS) is a significant and challenging problem in spectral graph theory. Recently, a simple criterion for almost controllable graphs to be DGS was proposed by Lin et al. (2026), utilizing the modified walk matrix. In this paper, we investigate the evolution of the modified walk matrix under disjoint union and join operations with a singleton vertex. We establish an exact algebraic identity for the determinant of the modified walk matrix of the resulting graph. Based on this identity and the DGS-criterion of Lin et al., we construct infinite families of almost controllable graphs that are DGS, extending the previous construction of Liu et al. (2019), which was restricted to controllable graphs.
