Zagreb, multiplicative Zagreb Indices and Coindices of ‎graphs

Subject Areas : International Journal of Industrial Mathematics

V. Ahmadi ¹ , M. R. ‎Darafsheh ² , J. ‎Hashemi‎ ³

1 - Department of Mathematics, Shahid Chamran University, Ahvaz, Iran.
2 - Department of Mathematics, Tehran University, Tehran, Iran.
3 - Department of Mathematics, Shahid Chamran University, Ahvaz, Iran.

Received: 2016-11-27 Accepted : 2016-11-27 Published : 2017-11-01

Keywords: Zagreb indices, Multiplicative Zagreb indices, Zagreb coindices, Multiplicative Zagreb coindice, ‎Link‎,

Abstract :

‎Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are respectivly defined by: $M_1(G)=\sum_{u\in V} d(u)^2, \hspace {.1 cm} M_2(G)=\sum_{uv\in E} d(u).d(v)$ and $ M_3(G)=\sum_{uv\in E}| d(u)-d(v)| $ , where d(u) is the degree of vertex u in G and uv is an edge of G connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of G are defined by: $PM_1(G)=\prod_{u\in V} d(u)^2$ and $PM_2(G)=\prod_{u\in V} d(u)^{d(u)}$. The first and second Zagreb coindices of G are defined by: $ \overline {M_1}(G) =\sum_{uv\notin E} ( d(u)+d(v))$ and $ \overline {M_2}(G) =\sum_{uv\notin E} d(u).d(v)$. The indices $ \overline {PM_1}(G) =\prod_{uv\notin E} d(u)+d(v)$ and $ \overline {PM_2}(G) =\prod_{uv\notin E} d(u).d(v)$ , are called the first and second multiplicative Zagreb coindices of G, respectively. In this article, we compute the first, second and third Zagreb indices and the first and second multiplicative Zagreb indices of some classes of dendrimers. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.Also, the multiplicative Zagreb indices are computed using link of ‎graphs.