# ON THE IRREGULARITY STRENGTH AND MODULAR IRREGULARITY STRENGTH OF FRIENDSHIP GRAPHS AND ITS DISJOINT UNION

### Abstract

For a simple, undirected graph G with, at most one isolated vertex and no isolated edges, a labeling *f:E(G)→{1,2,…,k _{1}}* of positive integers to the edges of

*G*is called irregular if the weights of each vertex of

*G*has a different value. The integer k

_{1}is then called the irregularity strength of

*G*. If the number of vertices in

*G*or the order of

*G*is

*|G|*, then the labeling

*μ:E(G)→{1,2,…,k*is called modular irregular if the remainder of the weights of each vertex of

_{2}}*G*divided by

*|G|*has a different value. The integer

*k*is then called the modular irregularity strength of

_{2}*G*. The disjoint union of two or more graphs, denoted by ‘+’, is an operation where the vertex and edge set of the result each be the disjoint union of the vertex and edge sets of the given graphs. This study discusses about the irregularity and modular irregularity strength of friendship graphs and some of its disjoint union, The result given is

*s(F*,

_{m}) = m + 1*ms(F*and

_{m}) = m + 1*ms(rF*, where

_{m}) = rm + ⌈r/2⌉*r*denotes the number of copies of friendship graphs

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*BAREKENG: J. Math. & App.*, vol. 16, no. 3, pp. 869-876, Sep. 2022.

Copyright (c) 2022 Fredrylo Alberth Noel Joddy Apituley, Mozart W. Talakua, Yopi Andry Lesnussa

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