Abstract
Abstract Recently, Todeschini et al. (Novel Molecular Structure Descriptors - Theory and Applications I, pp. 73-100, 2010), Todeschini and Consonni (MATCH Commun. Math. Comput. Chem. 64:359-372, 2010) have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: ∏ 1 = ∏ 1 ( G ) = ∏ v ∈ V ( G ) d G ( v ) 2 , ∏ 2 = ∏ 2 ( G ) = ∏ u v ∈ E ( G ) d G ( u ) d G ( v ) . These two graph invariants are called multiplicative Zagreb indices by Gutman (Bull. Soc. Math. Banja Luka 18:17-23, 2011). In this paper the upper bounds on the multiplicative Zagreb indices of the join, Cartesian product, corona product, composition and disjunction of graphs are derived and the indices are evaluated for some well-known graphs. MSC:05C05, 05C90, 05C07.
Highlights
Throughout this paper, we consider simple graphs which are finite, indirected graphs without loops and multiple edges
Suppose G is a graph with a vertex set V (G) and an edge set E(G)
For a graph G, the degree of a vertex v is the number of edges incident to v and is denoted by dG(v)
Summary
Throughout this paper, we consider simple graphs which are finite, indirected graphs without loops and multiple edges. For both connected graphs G and G , the equality holds in ( ) iff dG (ui)dG (ur) + n dG (ui) + dG (ur) + n = dG (ui)dG (uk) + n dG (ui) + dG (uk) + n for any uiur, uiuk ∈ E(G ); and dG (vj)dG (vr) + n dG (vj) + dG (vr) + n = dG (vj)dG (v ) + n dG (vj) + dG (v ) + n for any vjvr, vjv ∈ E(G ) as well as dG (ui)dG (vj) + n dG (vj) + n dG (ui) + n n = dG (ui)dG (v ) + n dG (v ) + n dG (ui) + n n for any ui ∈ V (G ), vj, v ∈ V (G ); and dG (ui)dG (vj) + n dG (vj) + n dG (ui) + n n = dG (uk)dG (vj) + n dG (vj) + n dG (uk) + n n for any vj ∈ V (G ), ui, uk ∈ V (G ) by Lemma .
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