Injective and Relative Injective Zagreb Indıces of Graphs

Authors : Akram ALQESMAH, Anwar ALWARDİ, R RANGARAJAN

Pages : 117-127

View : 13 | Download : 10

Publication Date : 2018-04-15

Article Type : Research Paper

Abstract :Let $G=insert ignore into journalissuearticles values(V,E);$ be a graph. The injective neighborhood of a vertex $u\in Vinsert ignore into journalissuearticles values(G);$ denoted by $N_{in}insert ignore into journalissuearticles values(u);$ is defined as $N_{in}insert ignore into journalissuearticles values(u);=\{v\in Vinsert ignore into journalissuearticles values(G);:|\Gammainsert ignore into journalissuearticles values(u,v);|\geq 1\}$, where $|\Gammainsert ignore into journalissuearticles values(u,v);|$ is the number of common neighborhoods between the vertices $u$ and $v$ in $G$. The cardinality of $N_{in}insert ignore into journalissuearticles values(u);$ is called the injective degree of the vertex $u$ in $G$ and denoted by $deg_{in}insert ignore into journalissuearticles values(u);$, \cite{20}. In this paper, we introduce the injective Zagreb indices of a graph $G$ as $M_1^{inj}insert ignore into journalissuearticles values(G);=\sum_{u\in Vinsert ignore into journalissuearticles values(G);}\big[deg_{in}insert ignore into journalissuearticles values(u);\big]^2$, $M_2^{inj}insert ignore into journalissuearticles values(G);=\sum_{uv\in Einsert ignore into journalissuearticles values(G);}deg_{in}insert ignore into journalissuearticles values(u);deg_{in}insert ignore into journalissuearticles values(v);$, respectively, and the relative injective Zagreb indices as $RM_1^{inj}insert ignore into journalissuearticles values(G);=\sum_{u\in Vinsert ignore into journalissuearticles values(G);}deg_{in}insert ignore into journalissuearticles values(u);deginsert ignore into journalissuearticles values(u);$, $RM_2^{inj}insert ignore into journalissuearticles values(G);=\sum_{uv\in Einsert ignore into journalissuearticles values(G);}\big[deg_{in}insert ignore into journalissuearticles values(u);deginsert ignore into journalissuearticles values(v);+deginsert ignore into journalissuearticles values(u);deg_{in}insert ignore into journalissuearticles values(v);\big]$, 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