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SOME RESULTS ON THE DIFFERENCE OF THE ZAGREB INDICES OF A GRAPH

Part of: Graph theory

Published online by Cambridge University Press:  02 June 2015

MINGQIANG AN  and
LIMING XIONG
MINGQIANG AN*
Affiliation:
College of Science, Tianjin University of Science and Technology, Tianjin300457, PR China email [email protected]
LIMING XIONG
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing100081, PR China email [email protected]
*
✉[email protected]
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Abstract

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The classical first and second Zagreb indices of a graph $G$ are defined as $M_{1}(G)=\sum _{v\in V(G)}d(v)^{2}$ and $M_{2}(G)=\sum _{e=uv\in E(G)}d(u)d(v),$ where $d(v)$ is the degree of the vertex $v$ of $G.$ Recently, Furtula et al. [‘On difference of Zagreb indices’, Discrete Appl. Math.178 (2014), 83–88] studied the difference of $M_{1}$ and $M_{2},$ and showed that this difference is closely related to the vertex-degree-based invariant $RM_{2}(G)=\sum _{e=uv\in E(G)}[d(u)-1][d(v)-1]$, the reduced second Zagreb index. In this paper, we present sharp bounds for the reduced second Zagreb index, given the matching number, independence number and vertex connectivity, and we also completely determine the extremal graphs.

Keywords

vertex degreeZagreb indexmatching numberindependence numbervertex connectivityextremal graphs

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C40: Connectivity 05C69: Dominating sets, independent sets, cliques 05C70: Factorization, matching, partitioning, covering and packing
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 177 - 186
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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