The spread of the spectrum of a graph

Gregory, David A.; Hershkowitz, Daniel; Kirkland, Steve

doi:10.1016/s0024-3795(00)00086-0

Cited by 60 publications

(59 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The spectral spread of unicyclic graphs has been studied in Shu and Wu (2003). Note that Conjecture 3.1 did appear in Gregory et al (2001), where it has been verified by computer for graphs up to 9 vertices, but remained unsolved. The conjecture is only about the structure of the extremal graphs.…”

Section: Theorem 24 Let G Be a Connected Graph On N Vertices With MImentioning

confidence: 99%

Open problems on graph eigenvalues studied with AutoGraphiX

Aouchiche

Caporossi

Hansen

2013

EURO Journal on Computational Optimization

View full text Add to dashboard Cite

Since the late forties of the last century, methods of operations research have been extensively used to solve problems in graph theory, and graph theory has been extensively used to model operations research problems and to solve optimization problems on graphs, e.g., shortest paths and network flow problems. More recently, methods of operations research and artificial intelligence have been used to advance graph theory per se, i.e., to find conjectures on graph theory invariants, to refute such conjectures and in some cases to find automated proofs or ideas of proofs. Among other systems, the AutoGraphiX system was developed since 1997 at GERAD (Montreal) by the present authors. Extensive experiments have been conducted which led to 1,700 conjectures, about 800 of which turned out to be easy and could be proved by the system, and about 600 further ones were proved by hand by us or graph theorists from various countries. Moreover, these results led to many generalizations and further papers. In this paper, we study four theoretical problems related to the eigenvalues of (the adjacency matrix of) a connected graph and to which AutoGraphiX was applied. Three of the problems are related to the maximum value of the irregularity, the maximum spectral spread and the upper bound of Nordhaus-Gaddum type on the index, over the class of connected graphs on n vertices. The fourth problem concerns the maximization of the energy (the sum of the absolute values of the eigenvalues) of a connected graph with fixed numbers of vertices and of cycles. We present a brief survey of the papers on or in connection with these problems, and give some new partial results.

show abstract

Section: Theorem 24 Let G Be a Connected Graph On N Vertices With MImentioning

confidence: 99%

Open problems on graph eigenvalues studied with AutoGraphiX

Aouchiche

Caporossi

Hansen

2013

EURO Journal on Computational Optimization

View full text Add to dashboard Cite

show abstract

“…Let G be a irregular graph with ∆ = δ + 1, then the bound (11) is better than the bound (9). And if ∆ = δ + 1, then the bound (11) is better than the bound (10).…”

Section: Theorem 45 ([23])mentioning

confidence: 99%

“…For details see the recent papers [10], [18], and the references quoted therein. It is well known that L is symmetric and positive semidefinite and µ n = 0.…”

Section: Introductionmentioning

confidence: 99%

The Laplacian spread of graphs

You

Liu

2012

Czech Math J

View full text Add to dashboard Cite

“…G has two distinct eigenvalues λ 1 and λ n , where λ 1 has multiplicity k and λ n has multiplicity n − k. Conversely, if λ 1 = λ 2 = · · · = λ k and λ k+1 = λ k+2 = · · · = λ n , then 2e = kλ 2 1 + (n − k)λ 2 n , which implies that λ n = − 2e−kλ 2 1 n−k , then the equality holds.…”

Section: If Equality Holds Between the First Two Expressions Then Thmentioning

confidence: 99%

“…In [2], the upper bound for regular graph is given. Here we will give some lower bounds with the same methods present in the former discussion.…”

Section: If Equality Holds Between the First Two Expressions Then Thmentioning

confidence: 99%