弦环图、多环图与混合图的结构和谱性质
Structural and Spectral Properties of Chordal Ring, Multi-ring and Mixed Graphs
AI总结：
本文推广弦环图，提出弦多环、弦环混合及弦多环混合图族，并利用平面镶嵌和阿贝尔群上的提升图方法，研究其最大顶点数、最小直径及邻接矩阵谱的闭式公式。
摘要：
弦环（CR）图是一类众所周知的图族，用于对计算机系统中所有节点均位于一个环上的某些互连网络进行建模。作为对CR图的推广，本文引入了弦多环（CMR）图、弦环混合（CRM）图以及弦多环混合（CMRM）图族。在混合图的情形中，我们可以同时具有边（无方向）和弧（有方向）。弦环图和弦环混合图是二部图且为3-正则图。它们由若干个（r≥1）无向或有向环以及连接这些环的一些边（弦）组成。特别地，对于CMR，当r=1时，即只有一个无向环时，我们得到已知的弦环图族。本文使用平面镶嵌来表示弦多环图。这使我们能够针对每个给定的直径获得其最大顶点数。此外，我们通过计算得到了任意顶点数取值下的最小直径。而且，当将其视为阿贝尔群上基图的提升图（也称为电压图）时，我们得到了谱的闭式公式，即其邻接矩阵的特征值多重集。
英文摘要：
The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number $r$ (for $r\geq 1$) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when $r=1$, that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we use plane tessellations to represent our chordal multi-ring graphs. This allows us to obtain their maximum number of vertices for every given diameter. Besides, we computationally obtain their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtain closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix.
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