Structural and Spectral Properties of Chordal Ring, Multi-ring and Mixed Graphs

TL;DR

This paper introduces new families of chordal multi-ring and mixed graphs, analyzing their structural, spectral, and diameter properties, with applications to interconnection network modeling.

Contribution

It generalizes chordal ring graphs to multi-ring and mixed variants, providing spectral formulas and diameter bounds using plane tessellations and algebraic methods.

Findings

Chordal multi-ring and mixed graphs are bipartite and 3-regular.

Maximum vertices for given diameter are determined via tessellations.

Closed-form spectra are derived using Abelian group lift graph models.

Abstract

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.