A survey of Nordhaus–Gaddum type relations

Aouchiche, Mustapha; Hansen, Pierre

doi:10.1016/j.dam.2011.12.018

Cited by 177 publications

(156 citation statements)

References 185 publications

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“…Finding an upper bound of Nordhaus-Gaddum type on the index over the class of connected graphs on n vertices, our third problem, was independently posed by Nosal (1970) and by Amin and Hakimi (1972). More than 40 years later, the problem remains open despite a series of improvements (see Aouchiche and Hansen 2012 for a survey) of the first bound given in Amin and Hakimi (1972) and Nosal (1970). The fourth problem concerns the maximization of the energy (the sum of the absolute values of the eigenvalues), as defined by Gutman (1978Gutman ( , 2001, of a connected graph with fixed numbers of vertices and of cycles.…”

Section: Introductionmentioning

confidence: 99%

“…For an exhaustive survey of such relations see Aouchiche and Hansen (2012) and the references therein. In the present paper, we are interested in NordhausGaddum relations only for the index.…”

Section: Nordhaus-gaddum Inequality For the Spectral Radiusmentioning

confidence: 99%

“…Conjecture 4.1 (Aouchiche et al 2008a;Aouchiche and Hansen 2012) For any simple graph G, with complement G, index λ 1 (G) and n vertices we have…”

Section: Nordhaus-gaddum Inequality For the Spectral Radiusmentioning

confidence: 99%

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Open problems on graph eigenvalues studied with AutoGraphiX

Aouchiche

Caporossi

Hansen

2013

EURO Journal on Computational Optimization

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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.

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Section: Introductionmentioning

confidence: 99%

Section: Nordhaus-gaddum Inequality For the Spectral Radiusmentioning

confidence: 99%

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Open problems on graph eigenvalues studied with AutoGraphiX

Aouchiche

Caporossi

Hansen

2013

EURO Journal on Computational Optimization

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“…An exhaustive survey is given in [2]. Here, we adduce only those closely related to the topic of our paper.…”

Section: Introductionmentioning

confidence: 99%

Improving upper bounds for the distinguishing index

Pilśniak

2017

Ars Math. Contemp.

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The distinguishing index of a graph G, denoted by D (G), is the least number of colours in an edge colouring of G not preserved by any non-trivial automorphism. We characterize all connected graphs G with D (G) ≥ ∆(G). We show that D (G) ≤ 2 if G is a traceable graph of order at least seven, and D (G) ≤ 3 if G is either claw-free or 3-connected and planar. We also investigate the Nordhaus-Gaddum type relation: 2 ≤ D (G) + D (G) ≤ max{∆(G), ∆(G)} + 2 and we confirm it for some classes of graphs.

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“…In graph theory this falls into the category of Nordhaus-Gaddum type problems ( [8], [2]). We show that although the number of lines in one hypergraph can be quite small, the arithmetic and geometric mean of the number of lines from both hypergraphs is much bigger.…”

mentioning

confidence: 99%

Nordhaus–Gaddum-type problems for lines in hypergraphs

Chen¹,

Miao

2016

Discrete Applied Mathematics

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We study the number of lines in hypergraphs in a more symmetric setting, where both the hypergraph and its complement are considered. In the general case and in some special cases, the lower bounds on the number of lines are much higher than their counterparts in single hypergraph setting or admit more elegant proofs. We show that the minimum value of product of the number of lines in both hypergraphs on n points is easily determined as n 2 ; and the minimum value of their sum is between Ω(n) and O(n log n). We also study some restricted classes of hypergraphs; and determine the tight bounds on the minimum sum when the hypergraph is derived from an Euclidean space, a real projective plane, or a tree.A hypergraph is an ordered pair (V, H) such that V is a set and H ⊆ 2 V is a family of subsets of V ; elements of V are the vertices of the hypergraph and members of H are its hyperedges; a hypergraph is called k-uniform if each of its hyperedges consists of k vertices; i.e. H ⊆ V k . In studying problems related to a De Bruijn-Erdős theorem, Chen and Chvátal [5] defined lines in 3-uniform hypergraphs. Specifically, for any u, v ∈ V , the line uv is defined as uv = {u, v} ∪ {p : {u, v, p} ∈ H} .

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A survey of Nordhaus–Gaddum type relations

Cited by 177 publications

References 185 publications

Open problems on graph eigenvalues studied with AutoGraphiX

Open problems on graph eigenvalues studied with AutoGraphiX

Improving upper bounds for the distinguishing index

Nordhaus–Gaddum-type problems for lines in hypergraphs

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