Journal
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY
Volume 92, Issue 2, Pages 177-186Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/S0004972715000386
Keywords
vertex degree; Zagreb index; matching number; independence number; vertex connectivity; extremal graphs
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Funding
- Natural Science Funds of China [11471037, 11171129]
- Specialised Research Fund for the Doctoral Program of Higher Education [20131101110048]
- Innovation Fund for Young Teachers of Tianjin University of Science and Technology [2014CXLG21]
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The classical first and second Zagreb indices of a graph G are defined as M-1(G) = Sigma(v is an element of V(G)) d(v)(2) and M-2(G) = Sigma(e) (=uv is an element of 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 RM2(G) = Sigma(e=uv is an element of 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.
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