INTEGRITY OF SOME DERIVED GRAPHS

By

B. Basavanagoud¹ and Shruti Policepatil²

Department of Mathematics, Karnatak University, Dharwad - 580 003, Karnataka, India

Email:¹b.basavanagoud@gmail.com, ²shrutipatil300@gmail.com

(Received : February 17, 2020 : In format: February 17, 2021; Revised in final form : June 21, 2021 )

DOI: https://doi.org/10.58250/Jnanabha.2021.51123

Abstract

One of the best parameter to measure the stability of a network is integrity as it takes into account both the amount of work done to damage the network and how badly the network is damaged. The integrity I(G) of a graph G is a measure of network vulnerability and is defined by I(G) = min{|S | + m(G − S )}, where S and m(G − S ) denote the subset of V and order of the largest component of G − S , respectively. In this paper, we study the integrity of line graph, jump graph, para-line graph of some standard graph families. In this way, we establish the relationship between integrity of basic graphs and integrity of their derived graphs. Also, we characterize few graphs having equal integrity values as that of derived graphs of same structured graphs. Further, we determine the integrity of generalized xyz-point-line transformation graphs Tˣʸ⁺ (G), Tˣʸ⁻(G), where x, y ∈ {0, 1, +, −} and Tˣʸ¹ (G), where x, y ∈ {0, 1, +, −}

2010 Mathematics Subject Classifications: 90C35, 05C40, 05C99.

Keywords and phrases: Vulnerability, integrity, line graph, jump graph, para-line graph, generalized xyz-point-line transformation graphs.

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