Injective and Relative Injective Zagreb Indıces of Graphs

Abstract

Let $G=(V,E)$ be a graph. The injective neighborhood of a vertex $u\in V(G)$ denoted by $N_{in}(u)$ is defined as $N_{in}(u)=\{v\in V(G):|\Gamma(u,v)|\geq 1\}$, where $|\Gamma(u,v)|$ is the number of common neighborhoods between the vertices $u$ and $v$ in $G$. The cardinality of $N_{in}(u)$ is called the injective degree of the vertex $u$ in $G$ and denoted by $deg_{in}(u)$, \cite{20}. In this paper, we introduce the injective Zagreb indices of a graph $G$ as $M_1^{inj}(G)=\sum_{u\in V(G)}\big[deg_{in}(u)\big]^2$, $M_2^{inj}(G)=\sum_{uv\in E(G)}deg_{in}(u)deg_{in}(v)$, respectively, and the relative injective Zagreb indices as $RM_1^{inj}(G)=\sum_{u\in V(G)}deg_{in}(u)deg(u)$, $RM_2^{inj}(G)=\sum_{uv\in E(G)}\big[deg_{in}(u)deg(v)+deg(u)deg_{in}(v)\big]$, respectively. Some properties of these topological indices are obtained. Exact values for some families of graphs and some graph operations are computed.

Keywords

• First injective Zagreb index,Second injective Zagreb index,First relative injective Zagreb index,Second relative injective Zagreb index

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