Conjectured Bounds For The Sum Of Squares Of Positive Eigenvalues Of A Graph

Keywords

05C50; Adjacency matrix; Hyper-energetic graphs; Inertia of a graph

Abstract

A well known upper bound for the spectral radius of a graph, due to Hong, is that μ21≤2m-n+1 if δ≥1. It is conjectured that for connected graphs n-1≤s+≤2m-n+1, where s+ denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete q-partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.

Publication Date

9-6-2016

Publication Title

Discrete Mathematics

Volume

339

Issue

9

Number of Pages

2215-2223

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.disc.2016.01.021

Socpus ID

84966325518 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84966325518

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