On the Roots of Domination Polynomials

Brown, Jason I.; Tufts, Julia

doi:10.1007/s00373-013-1306-z

Cited by 38 publications

(51 citation statements)

References 20 publications

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“…ATR roots were noted to have modulus at most 1 (in q) for small graphs, and it was conjectured in [2] that this was the case for all graphs. This contrasts sharply with what is known for other graph polynomials, such as chromatic polynomials [15], independence polynomials [5], and domination polynomials [6], where the roots are dense in the complex plane. Despite some results and generalizations in the affirmative [7,16], the conjecture for ATR roots was shown to be false in [14].…”

Section: Introductioncontrasting

confidence: 76%

“…, n − 1} gives spRel {u,v} (K − n ; q), and thus (8) Our examples of simple graphs with ATR roots outside of the unit disk are all of the form G (k,n) = G k,6k 3,3 [K − n (u, v)] for k ≥ 1 and 3 ≤ n ≤ 6. We start with the base graph G 1, 6 3,3 , replace every edge with a bundle of k edges, and then substitute the gadget K − n (u, v) for every edge. Before looking at particular examples we outline our general procedure for demonstrating that some G (k,n) has an ATR root outside of the unit disk.…”

Section: Simple Graphs With Atr Roots Outside Of the Unit Diskmentioning

confidence: 99%

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On the roots of all-terminal reliability polynomials

Brown

Mol

2017

Discrete Mathematics

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Given a graph G in which each edge fails independently with probability q ∈ [0, 1], the all-terminal reliability of G is the probability that all vertices of G can communicate with one another, that is, the probability that the operational edges span the graph. The all-terminal reliability is a polynomial in q whose roots (all-terminal reliability roots) were conjectured to have modulus at most 1 by Brown and Colbourn. Royle and Sokal proved the conjecture false, finding roots of modulus larger than 1 by a slim margin. Here, we present the first nontrivial upper bound on the modulus of any all-terminal reliability root, in terms of the number of vertices of the graph. We also find all-terminal reliability roots of larger modulus than any previously known. Finally, we consider the all-terminal reliability roots of simple graphs; we present the smallest known simple graph with all-terminal reliability roots of modulus greater than 1, and we find simple graphs with all-terminal reliability roots of modulus greater than 1 that have higher edge connectivity than any previously known examples.

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Section: Introductioncontrasting

confidence: 76%

Section: Simple Graphs With Atr Roots Outside Of the Unit Diskmentioning

confidence: 99%

On the roots of all-terminal reliability polynomials

Brown

Mol

2017

Discrete Mathematics

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“…Theorem 1 (see [15]). The domination polynomial of the star graph, ( 1, , ), where ∈ N, has a real root in the interval (−2 , − ln( )), for sufficiently large.…”

Section: Graphs With Few Domination Rootsmentioning

confidence: 99%

Graphs Whose Certain Polynomials Have Few Distinct Roots

Alikhani

2013

ISRN Discrete Mathematics

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Let= ( , ) be a simple graph. Graph polynomials are a well-developed area useful for analyzing properties of graphs. We consider domination polynomial, matching polynomial, and edge cover polynomial of G. Graphs which their polynomials have few roots can sometimes give surprising information about the structure of the graph. This paper is primarily a survey of graphs whose domination polynomial, matching polynomial, and edge cover polynomial have few distinct roots. In addition, some new unpublished results and questions are concluded.

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“…A polynomial that is more closely related to the independence polynomial is the edge cover polynomial and it was recently shown that its roots are bounded, in fact contained in the disk |z| < (2+ √ 3) 2 1+ √ 3 [11]. In contrast, the collection of all roots of independence polynomials [5], domination polynomials [8], and chromatic polynomials [28] are each dense in C.…”

Section: Introductionmentioning

confidence: 99%