On the roots of edge cover polynomials of graphs

Csíkvári, Péter; Oboudi, Mohammad Reza

doi:10.1016/j.ejc.2011.06.009

Cited by 24 publications

(26 citation statements)

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“…It is still believed that the reliability roots are bounded by some constant although the problem remains open. 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%

Maximum Modulus of Independence Roots of Graphs and Trees

Brown

Cameron

2020

Graphs and Combinatorics

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Section: Introductionmentioning

confidence: 99%

Maximum Modulus of Independence Roots of Graphs and Trees

Brown

Cameron

2020

Graphs and Combinatorics

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“…Theorem 32 (see [38]). Let be a connected graph whose edge cover polynomial has exactly two distinct roots.…”

Section: Theorem 31 (See [38]) Let Be a Graph Then ( ) = ( + 1) mentioning

confidence: 99%

“…Note that if graph has isolated vertex then we put ( , ) = 0 and if ( ) = ( ) = 0, then ( , ) = 1. For more detail on this polynomial refer to [37,38].…”

Section: Graphs With Few Edge-cover Rootsmentioning

confidence: 99%

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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 In addition, some new unpublished results and questions are concluded.

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On a Poset of Trees II

Csíkvári

2012

J. Graph Theory

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In this article, we study problems where one has to prove that certain graph parameter attains its maximum at the star and its minimum at the path among the trees on a fixed number of vertices. We give many applications of the so-called generalized tree shift which seems to be a powerful tool to attack the problems of the above-mentioned kind. We show that the generalized tree shift increases the largest eigenvalue of the adjacency matrix and Laplacian matrix, decreases the coefficients of the characteristic polynomials of these matrices in absolute value. We will prove similar theorems for the independence polynomial and the edge cover polynomial. The generalized tree shift induces a partially ordered set on trees having fixed number of vertices. The smallest element of this poset is the path, largest element is the star. Hence, the above-mentioned results imply the extremality of the path and the star for these parameters.

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On the roots of edge cover polynomials of graphs

Cited by 24 publications

References 9 publications

Maximum Modulus of Independence Roots of Graphs and Trees

Maximum Modulus of Independence Roots of Graphs and Trees

Graphs Whose Certain Polynomials Have Few Distinct Roots

On a Poset of Trees II

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