Written for mathematicians working with the theory of graph spectra, this (primarily theoretical) book presents relevant results considering the spectral properties of regular graphs. The book begins with a short introduction including necessary terminology and notation. The author then proceeds with basic properties, specific subclasses of regular graphs (like distance-regular graphs, strongly regular graphs, various designs or expanders) and determining particular regular graphs. Each chapter contains detailed proofs, discussions, comparisons, examples, exercises and also indicates possible applications. Finally, the author also includes some conjectures and open problems to promote further research.
Contents Spectral properties Particular types of regular graph Determinations of regular graphs Expanders Distance matrix of regular graphs
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Yes, you can access Regular Graphs by Zoran Stanić in PDF and/or ePUB format, as well as other popular books in Mathématiques & Mathématiques discrètes. We have over one million books available in our catalogue for you to explore.
Here we give a survey of the main graph-theoretic terminology, notation and necessary results. The presentation is separated into three sections. In the first two we deal with graph structure and give some observations including statistical data on regular graphs. In the third section we focus on the adjacency matrix and the corresponding spectrum.
Since all parts of this chapter can be found in numerous sources, by the assumption that the reader is familiar with most of the concepts presented, we orientate to a brief but very clear and intuitive exposition. More details are given in the introductory chapter of our previous book [290].
1.1Terminology and notation
Vertices and edges
Let G be a finite undirected simple graph (so, without loops or multiple edges). We denote its set of vertices (resp. edges) by V or V(G) (resp. E or E(G)). In addition, we assume that
. The quantities n = |V| and m = |E| are called the order and the size of G, respectively. Two vertices u and v are adjacent (or neighbours) if they are joined by an edge. In this case we write u ~ v and say that the edge uv is incident with vertices u and v. Similarly, two edges are adjacent if they are incident with a common vertex.
The set of neighbours (or the neighbourhood) of a vertex v is denoted N(v). The closed neighbourhood of v is denoted N[v] (= {v} ∪ N(v)).
Two graphs G and H are said to be isomorphic if there is a bijection between sets of their vertices which respects adjacencies. If so, then we write G ≅ H. Observe that the graphs illustrated in Figure 1.1 are isomorphic. In particular, an automorphism of a graph is an isomorphism to itself.
The degree dv of a vertex v is the number of edges incident with it. The minimal and the maximal vertex degrees in a graph are denoted δ and Δ, respectively. A vertex of degree 1 is called an endvertex or a pendant vertex.
We say that a graph G is regular of degree r if all its vertices have degree r. If so, then G is refe...
