Topic
Triangle-free graph
About: Triangle-free graph is a research topic. Over the lifetime, 830 publications have been published within this topic receiving 18453 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: This paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to $(m + n)\sqrt n $.
Abstract: The present paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to $(m + n)\sqrt n $.
3,014 citations
TL;DR: The proof exploits the characteristic roots and vectors of the adjacency matrix (and its principal submatrices) of the graph to prove the existence of connected, undirected graphs homogeneous of degree d and of diameter k.
Abstract: This note treats the existence of connected, undirected graphs homogeneous of degree d and of diameter k, having a number of nodes which is maximal according to a certain definition. For k = 2 unique graphs exist for d = 2, 3, 7 and possibly for d = 57 (which is undecided), but for no other degree. For k = 3 a graph exists only for d = 2. The proof exploits the characteristic roots and vectors of the adjacency matrix (and its principal submatrices) of the graph.
593 citations
1 Jan 2004
TL;DR: Cameron et al. as mentioned in this paper introduced the concept of graph Laplacians and showed that they can be used to construct strong regular graphs and distance-transitive graphs, including Cayley graphs.
Abstract: Foreword Peter J. Cameron Introduction 1. Eigenvalues of graphs Michael Doob 2. Graphs and matrices Richard A. Brualdi and Bryan L. Shader 3. Spectral graph theory Dragos Cvetkovic and Peter Rowlinson 4. Graph Laplacians Bojan Mohar 5. Automorphism groups Peter J. Cameron 6. Cayley graphs Brian Alspach 7. Finite symmetric graphs Cheryle E. Praeger 8. Strongly regular graphs Peter J. Cameron 9. Distance-transitive graphs Arjeh M. Cohen 10. Computing with graphs and groups Leonard H. Soicher.
300 citations
TL;DR: It is shown that for every e > 0 there is a natural number r such that in almost every r-regular graph of order n, every set of u ≤ n/2 vertices is joined by at least (r/2 - e)u edges to the rest of the graph.
264 citations
TL;DR: In this article, a bipartite, q-regular, edge-transitive graph CD(k, q) of order v ≤ 2qk−t+1 and girth g ≥ k + 5 was constructed.
Abstract: Let k ≥ 1 be an odd integer, t = b k+2 4 c, and q be a prime power. We construct a bipartite, q–regular, edge–transitive graph CD(k, q) of order v ≤ 2qk−t+1 and girth g ≥ k + 5. If e is the the number of edges of CD(k, q), then e = Ω(v 1 k−t+1 ). These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g ≥ 5, g 6= 11, 12. For g ≥ 24, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for 5 ≤ g ≤ 23, g 6= 11, 12, it improves on or ties existing bounds.
262 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2023 | 2 |
| 2022 | 3 |
| 2021 | 2 |
| 2020 | 7 |
| 2019 | 3 |
| 2018 | 6 |