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.

Full text (pdf)

Opuscula Mathematica - cover

Cite this article as:
Kexiang Xu, Kinkar Ch. Das, Kechao Tang, On the multiplicative Zagreb coindex of graphs, Opuscula Math. 33, no. 1 (2013), 191-204, http://dx.doi.org/10.7494/OpMath.2013.33.1.191

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.