Opuscula Math. 33, no. 1 (2013), 191-204
http://dx.doi.org/10.7494/OpMath.2013.33.1.191

Opuscula Mathematica

# On the multiplicative Zagreb coindex of graphs

Kexiang Xu
Kinkar Ch. Das
Kechao Tang

Abstract. For a (molecular) graph $$G$$ with vertex set $$V(G)$$ and edge set $$E(G)$$, the first and second Zagreb indices of $$G$$ are defined as $$M_1(G) = \sum_{v \in V(G)} d_G(v)^2$$ and $$M_2(G) = \sum_{uv \in E(G)} d_G(u)d_G(v)$$, respectively, where $$d_G(v)$$ is the degree of vertex $$v$$ in $$G$$. The alternative expression of $$M_1(G)$$ is $$\sum_{uv \in E(G)}(d_G(u) + d_G(v))$$. Recently Ashrafi, Došlić and Hamzeh introduced two related graphical invariants $$\overline{M}_1(G) = \sum_{uv \notin E(G)}(d_G(u)+d_G(v))$$ and $$\overline{M}_2(G) = \sum_{uv \notin E(G)} d_G(u)d_G(v)$$ named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants $$\overline{\Pi}_1(G) = \prod_{uv \notin E(G)}(d_G(u)+d_G(v))$$ and $$\overline{\Pi}_2(G) = \prod_{uv \notin E(G)} d_G(u)d_G(v)$$ as the respective multiplicative versions of $$\overline{M}_i$$ for $$i = 1, 2$$. In this paper, we have reported some properties, especially upper and lower bounds, for these two graph invariants of connected (molecular) graphs. Moreover, some corresponding extremal graphs have been characterized with respect to these two indices.

Keywords: vertex degree, tree, upper or lower bound.

Mathematics Subject Classification: 05C05, 05C07, 05C35.

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• Kexiang Xu
• Nanjing University of Aeronautics and Astronautics, College of Science, Nanjing, Jiangsu 210016, PR China
• Kinkar Ch. Das
• Sungyunkwan University, Department of Mathematics, Suwon 440-746, Republic of Korea
• Kechao Tang
• Nanjing University of Aeronautics and Astronautics, College of Science, Nanjing, Jiangsu 210016, PR China
• Revised: 2012-06-08.
• Accepted: 2012-06-11.