Topic
Circulant graph
About: Circulant graph is a research topic. Over the lifetime, 1556 publications have been published within this topic receiving 28865 citations.
Papers published on a yearly basis
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Papers
TL;DR: Two backtracking algorithms are presented, using a branchand-bound technique [4] to cut off branches that cannot lead to a clique, and generates cliques in a rather unpredictable order in an attempt to minimize the number of branches to be traversed.
Abstract: Description bttroductian. A maximal complete subgraph (clique) is a complete subgraph that is not contained in any other complete subgraph. A recent paper [1] describes a number of techniques to find maximal complete subgraphs of a given undirected graph. In this paper, we present two backtracking algorithms, using a branchand-bound technique [4] to cut off branches that cannot lead to a clique. The first version is a straightforward implementation of the basic algorithm. It is mainly presented to illustrate the method used. This version generates cliques in alphabetic (lexicographic) order. The second version is derived from the first and generates cliques in a rather unpredictable order in an attempt to minimize the number of branches to be traversed. This version tends to produce the larger cliques first and to generate sequentially cliques having a large common intersection. The detailed algorithm for version 2 is presented here. Description o f the algorithm--Version 1. Three sets play an important role in the algorithm. (1) The set compsub is the set to be extended by a new point or shrunk by one point on traveling along a branch of the backtracking tree. The points that are eligible to extend compsub, i.e. that are connected to all points in compsub, are collected recursively in the remaining two sets. (2) The set candidates is the set of all points that will in due time serve as an extension to the present configuration of compsub. (3) The set not is the set of all points that have at an earlier stage already served as an extension of the present configuration of compsub and are now explicitly excluded. The reason for maintaining this set trot will soon be made clear. The core of the algorithm consists of a recursively defined extension operator that will be applied to the three sets Just described. It has the duty to generate all extensions of the given configuration of compsub that it can make with the given set of candidates and that do not contain any of the points in not. To put it differently: all extensions of compsub containing any point in not have already been generated. The basic mechanism now consists of the following five steps:
2,696 citations
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TL;DR: In this article, the maximum number of cliques possible in a graph with n nodes is determined and bounds are obtained for the number of different sizes of clique possible in such a graph.
Abstract: A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph withn nodes is determined. Also, bounds are obtained for the number of different sizes of cliques possible in such a graph.
1,002 citations
TL;DR: The graph theoretic properties of this variant of the domination number of a graph G, a function f : V→{0,1,2} satisfying the condition that every vertex u is adjacent to at least one vertex v for which f(v)=2, are studied.
601 citations
597 citations
TL;DR: In this paper, the Laplacian matrix of a finite undirected graph G with no loops or multiple edges is defined and the structure of the graph G is related to the eigenvalues of A(G): in particular, it is shown that all eigen values of Δ(G) are non-negative, less than or equal to the number of vertices, and less than/or equal to twice the maximum vertex degree.
Abstract: Let G be a finite undirected graph with no loops or multiple edges. We define the Laplacian matrix of G,Δ(G)by Δij= degree of vertex i and Δij−1 if there is an edge between vertex i and vertex j. In this paper we relate the structure of the graph G to the eigenvalues of A(G): in particular we prove that all the eigenvalues of Δ(G) are non-negative, less than or equal to the number of vertices, and less than or equal to twice the maximum vertex degree. Precise conditions for equality are given.
560 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 5 |
| 2024 | 7 |
| 2023 | 16 |
| 2022 | 27 |
| 2021 | 24 |
| 2020 | 13 |