Finding extremal total irregularity of polyomino chain by transformation method

Subject Areas : Statistics

Zahra  Yarahmadi ¹ (Department of Mathematics, Faculty of Science, Khorramabad Branch, Islamic Azad University, Khorramabad, I. R. Iran)

Received: 2020-10-08 Accepted : 2021-01-23 Published : 2021-05-22

Keywords: انتقال, زنجیر زیگزاگ, زنجیر خطی, زنجیر پولیومینو, بی‌نظمی کامل,

Abstract :

Let be a simple and undirected graph, consists of a set of vertices and a set of edges If the vertices and are connected by an edge then we write . For any vertices , the degree of a vertex in , denoted by , is the number of edges of incident with . A topological index is a numerical quantity related to a graph which is invariant under graph automorphisms, Let be a topological index of a graph , for every graph isomorphic to , we have . The first of topological indices, which based on degree of vertices of graphs are first and second Zagreb indices. In this paper we study on another kind of this graph invariants called total irregularity. The total irregularity of G is a graph invariant and defined as the follows, . In this paper, we introduce two kind of transformations on polyomino chains and by using these transformations obtain upper and lower bounds of the total irregularity of polyomino chains. Moreover, we prove that the linear chain and zigzag chain are extremal polyomino chains with respect to total irregularity.

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