# The Edge Cover Probability Polynomial of a Graph and Optimal Network Construction

Research output: Contribution to journalArticle

### Abstract

Given a uniform probability <formula><tex>$\rho, 0 &lt; \rho &lt; 1$</tex></formula>, of selecting edges independently from a graph <formula><tex>$G$</tex></formula>, we define the edge cover probability polynomial <formula><tex>$Ep(G, \rho)$</tex></formula> of <formula><tex>$G$</tex></formula> to be the probability of randomly selecting an edge cover of <formula><tex>$G$</tex></formula>. We provide general, and in some cases specific, formulas for obtaining <formula><tex>$Ep(G, \rho)$</tex></formula>. We then demonstrate the existence of graphs which have either the largest or the smallest <formula><tex>$Ep(G, \rho)$</tex></formula> within its class for all &#x03C1;. The classes we consider are trees, unicyclic graphs, and connected graphs having one more edge than the number of vertices. Thus we determine the optimal constructions with respect to edge covers within the context of these classes.

Original language English (US) IEEE Transactions on Network Science and Engineering https://doi.org/10.1109/TNSE.2018.2820062 Accepted/In press - Mar 26 2018

Polynomials

### All Science Journal Classification (ASJC) codes

• Control and Systems Engineering
• Computer Science Applications
• Computer Networks and Communications

### Cite this

title = "The Edge Cover Probability Polynomial of a Graph and Optimal Network Construction",
abstract = "Given a uniform probability $\rho, 0 < \rho < 1$, of selecting edges independently from a graph $G$, we define the edge cover probability polynomial $Ep(G, \rho)$ of $G$ to be the probability of randomly selecting an edge cover of $G$. We provide general, and in some cases specific, formulas for obtaining $Ep(G, \rho)$. We then demonstrate the existence of graphs which have either the largest or the smallest $Ep(G, \rho)$ within its class for all ρ. The classes we consider are trees, unicyclic graphs, and connected graphs having one more edge than the number of vertices. Thus we determine the optimal constructions with respect to edge covers within the context of these classes.",
author = "Yatauro, {Michael Robert}",
year = "2018",
month = "3",
day = "26",
doi = "10.1109/TNSE.2018.2820062",
language = "English (US)",
journal = "IEEE Transactions on Network Science and Engineering",
issn = "2327-4697",
publisher = "IEEE Computer Society",

}

In: IEEE Transactions on Network Science and Engineering, 26.03.2018.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The Edge Cover Probability Polynomial of a Graph and Optimal Network Construction

AU - Yatauro, Michael Robert

PY - 2018/3/26

Y1 - 2018/3/26

N2 - Given a uniform probability $\rho, 0 < \rho < 1$, of selecting edges independently from a graph $G$, we define the edge cover probability polynomial $Ep(G, \rho)$ of $G$ to be the probability of randomly selecting an edge cover of $G$. We provide general, and in some cases specific, formulas for obtaining $Ep(G, \rho)$. We then demonstrate the existence of graphs which have either the largest or the smallest $Ep(G, \rho)$ within its class for all ρ. The classes we consider are trees, unicyclic graphs, and connected graphs having one more edge than the number of vertices. Thus we determine the optimal constructions with respect to edge covers within the context of these classes.

AB - Given a uniform probability $\rho, 0 < \rho < 1$, of selecting edges independently from a graph $G$, we define the edge cover probability polynomial $Ep(G, \rho)$ of $G$ to be the probability of randomly selecting an edge cover of $G$. We provide general, and in some cases specific, formulas for obtaining $Ep(G, \rho)$. We then demonstrate the existence of graphs which have either the largest or the smallest $Ep(G, \rho)$ within its class for all ρ. The classes we consider are trees, unicyclic graphs, and connected graphs having one more edge than the number of vertices. Thus we determine the optimal constructions with respect to edge covers within the context of these classes.

UR - http://www.scopus.com/inward/record.url?scp=85044854391&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044854391&partnerID=8YFLogxK

U2 - 10.1109/TNSE.2018.2820062

DO - 10.1109/TNSE.2018.2820062

M3 - Article

AN - SCOPUS:85044854391

JO - IEEE Transactions on Network Science and Engineering

JF - IEEE Transactions on Network Science and Engineering

SN - 2327-4697

ER -