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Eigenvalues and triangles in graphs

Part of: Graph theory

Published online by Cambridge University Press:  28 September 2020

Huiqiu Lin ,
Bo Ning  and
Baoyindureng Wu
Huiqiu Lin
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai200237, PR China
Bo Ning*
Affiliation:
College of Computer Science, Nankai University, Tianjin300071, PR China
Baoyindureng Wu
Affiliation:
College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang830046, PR China
*
*Corresponding author. Email: bo.ning@nankai.edu.cn
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Abstract

Bollobás 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 ${\rm{\lambda }}_1^2(G) + {\rm{\lambda }}_2^2(G) \le (r - 1)/r \cdot 2m$, where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the case r=2, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdös and Nosal respectively, we prove that every non-bipartite graph of order and size contains a triangle if one of the following is true: (i) ${{\rm{\lambda }}_1}(G) \ge \sqrt {m - 1} $ and $G \ne {C_5} \cup (n - 5){K_1}$, and (ii) ${{\rm{\lambda }}_1}(G) \ge {{\rm{\lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$ and $G \ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$, where $S({K_{[(n - 1)/2],[(n - 1)/2]}})$ is obtained from ${K_{[(n - 1)/2],[(n - 1)/2]}}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.

MSC classification

Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 2 , March 2021 , pp. 258 - 270
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Research supported by NSFC (grants 11771141 and 12011530064).

‡

This work is supported by NSFC (No. 11971346).

§

Research supported by NSFC (grant 11571294).

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