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

On the multiplicative Zagreb coindex of graphs




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.
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.

RSS Feed

horizontal rule

ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

horizontal rule

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.