Eigenvalues and triangles in graphs
TL;DR: It is proved that every non-bipartite graph of order and size contains a triangle if one of the following is true: $(G) \ge \sqrt {m - 1} $ and $G \ne {C_5} \cup (n - 5){K_1}$.
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Abstract: Bollobas and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.
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Spectral extrema of graphs with fixed size: Cycles and complete bipartite graphs
TL;DR: It is shown that if ρ ( G ) ≥ m, then G contains K 2, r + 1 unless G is a star; and G contains either C 3 + or C 4 + unless GIs a complete bipartite graph, where C t + denotes the graph obtained from C t and C 3 by identifying an edge.
67
Adjacency eigenvalues of graphs without short odd cycles
01 Jan 2022
TL;DR: In this paper , an upper bound on λ12k+λ22k of n-vertex {C3,C5,…,C2k+1}-free graphs is established, where k is a positive integer.
27
A spectral version of Mantel's theorem
TL;DR: In this article , the authors show that every non-bipartite graph G of size m with ρ (G ) ≥ ρ ⁎ ( m ) contains a triangle unless G ≅ S K 2 , m − 1 2 , where ρ is the largest root of x 3 − x 2 − (m − 2 ) x + ( m − 3 ) = 0.
23
A spectral condition for odd cycles in non-bipartite graphs
Huiqiu Lin,Hangtian Guo +1 more
TL;DR: In this article, it was shown that e x s p ( n, C 2 k + 3 ) = ρ ( S 2 k − 1 ( K s, t ), where k ≥ 2.
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A survey on spectral conditions for some extremal graph problems
Yongtao Li,Weijun Liu,Lihua Feng +2 more
TL;DR: In this paper, the spectral extremal results on the Tur\'{a}n type problems in graph theory are summarized in terms of the adjacency spectral radius and the signless Laplacian spectral radius for various graphs.
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References
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Graph Theory
J. A. Bondy,U.S.R Murty +1 more
- 14 Aug 2008
TL;DR: This book provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal, and is suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science.
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Spectra of graphs : theory and application
Dragoš Cvetković,Michael Doob,Horst Sachs +2 more
- 01 Jan 1995
TL;DR: The Spectrum and the Group of Automorphisms as discussed by the authors have been used extensively in Graph Spectra Techniques in Graph Theory and Combinatory Applications in Chemistry an Physics. But they have not yet been applied to Graph Spectral Biblgraphy.
2.2K
Eigen values and expanders
Noga Alon,Noga Alon +1 more
TL;DR: It is shown that a regular bipartite graph is an expanderif and only if the second largest eigenvalue of its adjacency matrix is well separated from the first.
1.3K
Maxima for Graphs and a New Proof of a Theorem of Turán
T. S. Motzkin,E. G. Straus +1 more
TL;DR: In this article, the maximum of a square-free quadratic form on a simplex was investigated and the following question was suggested by a problem of J. E. MacDonald Jr.
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