Yıl 2018, Cilt 6, Sayı 1, Sayfalar 117 - 127 2018-04-15
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Injective and Relative Injective Zagreb Indıces of Graphs

Akram Alqesmah [1] , Anwar Alwardi [2] , R. Rangarajan [3]

49 110

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.
First injective Zagreb index, Second injective Zagreb index, First relative injective Zagreb index, Second relative injective Zagreb index
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