On Two Conjectures of Spectral Graph Theory

Original Paper
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Abstract

Let \(G=(V,\,E)\) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix and the signless Laplacian matrix of G are \(L(G)=D(G)-A(G)\) and \(Q(G)=D(G)+A(G)\), respectively. Also denote by \(\lambda _1(G)\), a(G), \(q_1(G)\) and \(\delta (G)\) the largest eigenvalue of A(G), the second smallest eigenvalue of L(G), the largest eigenvalue of Q(G) and the minimum degree of G, respectively. In this paper, we give partial proofs to the following two conjectures:
(i)

Aouchiche (Comparaison Automatisée d’Invariants en Théorie des Graphes, 2006) if G is a connected graph, then \(a(G)/\delta (G)\) is minimum for graph composed of 2 triangles linked with a path.

(ii)

Aouchiche et al. (Linear Algebra Appl 432:2293–2322, 2010) and Cvetković et al. (Publ Inst Math Beogr 81(95):11–27, 2007) if G is a connected graph with \(n\ge 4\) vertices, then \(q_1(G)-2\,\lambda _1(G)\le n-2\,\sqrt{n-1}\) with equality holding if and only if \(G\cong K_{1,n-1}\).

Keywords

Graph Spectral radius Signless Laplacian spectral radius Algebraic connectivity 

Mathematics Subject Classification

05C50 15A18 15A36 

Notes

Acknowledgements

The authors are grateful to the anonymous referee for helpful suggestions and valuable comments, which led to an improvement of the original manuscript. The first author is supported by the Sungkyun research fund, Sungkyunkwan University, 2017, and National Research Foundation of the Korean government with Grant no. 2017R1D1A1B03028642. The second author is partially supported by NSFC (No. 11571123), the Training Program for Outstanding Young Teachers in University of Guangdong Province (No. YQ2015027) and Guangdong Engineering Research Center for Data Science (No. 2017A-KF02).

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Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsSungkyunkwan UniversitySuwonRepublic of Korea
  2. 2.Department of Applied MathematicsSouth China Agricultural UniversityGuangzhouPeople’s Republic of China
  3. 3.College of Mathematics and StatisticsShenzhen UniversityShenzhenPeople’s Republic of China

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