Optimal Domination Polynomials

Beaton, Iain; Brown, Jason I.; Cox, Danielle

doi:10.1007/s00373-020-02202-8

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…Therefore, we rely on the analysis of the roots of a related polynomial and a Möbius transformation to show U n is more reliable than C n . See [12] for further details on Möbius transformations in general, [7,9] for use of Möbius transformations for roots of graph polynomials, and [2] for their use in finding UMR graphs with respect to other notions of reliability. Proof.…”

Section: Proofmentioning

confidence: 99%

“…Let G be a graph where each node is operational independently with probability p and let t, s ∈ V(G) be target nodes. Then the two-terminal node reliability of G is the probability that the graph is connected and contains t and s. In a very recent paper [6], it was shown that for every n and m there exists a UMR graph with respect to two-terminal node reliability; however when the target nodes are required to have distance at least 3 from each other, there exists a UMR graph on n nodes and m edges if and only if m ≤ 8, m ≥ ⌊ (n−1) 2 4…”

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The node cop‐win reliability of unicyclic and bicyclic graphs

Ahmed

Cameron

2021

Networks

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Various models to quantify the reliability of a network have been studied where certain components of the graph may fail at random and the probability that the remaining graph is connected is the proxy for reliability. We introduce a strengthening of one of these models by considering the probability that the remaining graph is cop-win. A graph is cop-win if one cop can win the game Cops and Robber. More precisely, for a graph G with nodes that are operational independently with probability p, the node cop-win reliability of G, denoted NCRel(G, p), is the probability that the operational nodes induce a cop-win subgraph of G. It is then of interest to find graphs G with n nodes and m edges such that NCRel(G, p) ≥ NCRel(H, p) for all p ∈ [0, 1] and all graphs H with n nodes and m edges. We show that such graphs exist among unicyclic and bicyclic graphs, respectively.

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The node cop‐win reliability of unicyclic and bicyclic graphs

Ahmed

Cameron

2021

Networks

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A rapid numerical method for nonlinear generalized time-fractional kawahara equations via domination polynomials of complete graph

Nirmala,

Kumbinarasaiah

2024

Phys. Scr.

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The key objective of this study is to examine the nonlinear Time-fractional generalized Kawahara equation (N-TF-GKE), a significant nonlinear equation that appears in a variety of physical domains, including ion-acoustic waves in plasmas, shallow water waves, and nonlinear acoustics. This Equation exhibits complex dynamic properties, most notably the appearance of shock waves, solitary waves, and chaotic solutions. We use a graph theoretic polynomial approach to obtain numerical solutions of the N-TF-GKE. We present the dominance polynomials of complete graphs (DPC), a new class of graph polynomials, to develop an efficient operational integration matrix to approximate the N-TF-GKE. The N-TF-GKE transformed into a collection of nonlinear algebraic equations with the standard collocation points. We acquire the numerical solutions for the Domination polynomial using Newton's approach. We validate the efficiency and robustness of our domination polynomial collocation method (DCM) with a set of numerical experiments and comparative analyses.

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On the Roots of a Family of Polynomials

Jianu

2023

Fractal Fract

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The aim of this paper is to give a characterization of the set of roots of a special family of polynomials. This family is relevant in reliability theory since it contains the reliability polynomials of the networks created by series-parallel compositions. We prove that the set of roots is bounded, being contained in the two disks of the radius equal to the golden ratio, centered at 0 and at 1. We study the closure of the set of roots and prove that it includes two disks centered at 0 and 1 of a radius slightly greater than 1, as well as the sinusoidal spirals centered at 0 and at 1, respectively. The expression of some limit points is also provided.

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Optimal Domination Polynomials

Cited by 3 publications

References 11 publications

The node cop‐win reliability of unicyclic and bicyclic graphs

The node cop‐win reliability of unicyclic and bicyclic graphs

A rapid numerical method for nonlinear generalized time-fractional kawahara equations via domination polynomials of complete graph

On the Roots of a Family of Polynomials

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