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On the degree distance of unicyclic graphs with given matching number. (English) Zbl 1328.05060
Summary: Let \(G=(V_G,E_G)\) be a connected graph. The degree distance of \(G\) is defined as \(D'(G)=\sum_{\{u,v\}\subseteq V_G}(d_G(v)+d_G(v))d_G(u,v)\), where \(d_G(v)\) is the degree of vertex \(v,d_G(u,v)\) denotes the distance between \(u\) and \(v\) in \(G\). This parameter was introduced, independently, by A. A. Dobrynin and A. A. Kochetova [“Degree distance of a graph: a degree analogue of the Wiener index”, J. Chem. Inf. Comput. Sci. 34, No. 5, 1082–1086 (1994; doi:10.1021/ci00021a008)] and by I. Gutman [“Selected properties of the Schultz molecular topological index”, ibid. 34, No. 5, 1087–1089 (1994; doi:10.1021/ci00021a009)] as a weighted version of the Wiener index. L. Feng et al. [Graphs Comb. 29, No. 3, 449–462 (2013; Zbl 1267.05107)] characterized \(n\)-vertex unicyclic graphs with given matching number having the minimal degree distance. As a continuance of it, in this paper the \(n\)-vertex unicyclic graphs of given matching number with the second and third minimal degree distance are identified respectively.

MSC:
05C12 Distance in graphs
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
Software:
nauty
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