高度对称图的对称实现与谱实现
Symmetric and Spectral Realizations of Highly Symmetric Graphs
摘要：
图$G=(V,E)$的实现是将$d$维欧几里得空间中的点分配给每个顶点的映射$v\colon V\to\Bbb R^d$。我们从表示论（表示某些对称性）、谱图论（满足某些自应力条件）和刚性理论（承认不改变对称性的变形）的角度研究图的实现。我们探讨这些观点之间的联系，重点是实现高度对称的图形（弧传递/距离传递）和多少对称性是必要的，以确保实现是平衡的，光谱，刚性等问题，我们包括许多例子，给一个广泛的概述对称和光谱图实现的可能性和限制。
英文摘要：
A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain symmetries), spectral graph theory (satisfying certain self-stress conditions) and rigidity theory (admitting deformations that do not alter the symmetry properties). We explore the connections between these perspectives, with a focus on realizations of highly symmetric graphs (arc-transitive/distance-transitive) and the question of how much symmetry is necessary to ensure that a realization is balanced, spectral, rigid etc. We include many examples to give a broad overview of the possibilities and restrictions of symmetric and spectral graph realizations.
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