General multiplicative Zagreb indices of trees and unicyclic graphs with given matching number
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General multiplicative Zagreb indices of trees and unicyclic graphs with given matching number Tomáš Vetrík1
· Selvaraj Balachandran1,2
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The Zagreb index of a graph G is defined as P1a (G) = first general multiplicative a (degG (v)) and the second general multiplicative Zagreb index is P2a (G) = v∈V (G) a degG (v) , where V (G) is the vertex set of G, deg (v) is the degree G v∈V (G) (degG (v)) of v in G and a = 0 is a real number. We present lower and upper bounds on the general multiplicative Zagreb indices for trees and unicyclic graphs of given order with a perfect matching. We also obtain lower and upper bounds for trees and unicyclic graphs of given order and matching number. All the trees and unicyclic graphs which achieve the bounds are presented, thus our bounds are sharp. Bounds for the classical multiplicative Zagreb indices are special cases of our theorems and those bounds are new results as well. Keywords Tree · Unicyclic graph · Multiplicative Zagreb index · Matching
1 Introduction Let G be a simple connected graph with vertex set V (G) and edge set E(G). The order n of a graph G is the number of vertices of G. The degree of a vertex v ∈ V (G), denoted by degG (v), is the number of edges incident to v. A pendant vertex is a vertex of degree 1. A pendant path of G is a path, in which one terminal vertex is of degree at least 3 in G, the other terminal vertex is a pendant vertex, and every internal vertex
The work of T. Vetrík is based on the research supported by the National Research Foundation of South Africa (Grant Number 126894).
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Tomáš Vetrík [email protected]
1
Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa
2
Department of Mathematics, School of Arts, Sciences and Humanities, SASTRA Deemed University, Thanjavur, India
123
Journal of Combinatorial Optimization
(if any exists) has degree 2 in G. We denote the path, the cycle and the star having n vertices by Pn , Cn and Sn , respectively. A tree is a connected graph containing no cycles and a unicyclic graph is a connected graph with exactly one cycle. The girth of a unicyclic graph is the length of its cycle. The distance dG (u, v) between two vertices u, v ∈ V (G) is the number of edges in a shortest path connecting them. For a set S ⊆ V (G), dG (u, S) is the distance between u ∈ V (G) and a vertex of S which is closest to u in G. The eccentricity of u, eccG (u), is the distance between u and any vertex furthest from u in G. A matching is a set of edges of G such that no two edges have a vertex in common. A perfect matching of G is a matching in which every vertex of G is incident to exactly one edge of the matching. The matching number of a graph is the size of a maximum independent edge set. Topological indices play a significant role in chemistry, materials science, pharmaceutical sciences and engineering, since they can be correlated with a large number of physical and chemical propertie
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