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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.)
##### Keywords:
degree distance; matching number; unicyclic graph
nauty
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##### References:
 [1] Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan Press, New York (1976) · Zbl 1226.05083 [2] Bucicovschi, O; Cioabǎ, SM, The minimum degree distance of graphs of given order and size, Discrete Appl. Math., 156, 3518-3521, (2008) · Zbl 1168.05308 [3] Chang, A; Tian, F, On the spectral radius of unicyclic graphs with perfect matching, Linear Algebra Appl., 370, 237-250, (2003) · Zbl 1030.05074 [4] Dobrynin, AA; Kochetova, AA, Degree distance of a graph: a degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci., 34, 1082-1086, (1994) [5] Dankelmann, P; Gutman, I; Mukwembi, S; Swart, HC, On the degree distance of a graph, Discrete Appl. Math., 157, 2773-2777, (2009) · Zbl 1209.05069 [6] Diudea, M.V., Gutman, I., Jäntschi, L.: Molecular Topology. Nova, Huntington (2001) [7] Dobrynin, A; Entringer, R; Gutman, I, Wiener index of trees: theory and applications, Acta Appl. Math., 66, 211-249, (2001) · Zbl 0982.05044 [8] Feng, LH; Liu, WJ; Ilić, A; Yu, GH, Degree distance of unicyclic graphs with given matching number, Graphs Comb., 29, 449-462, (2013) · Zbl 1267.05107 [9] Gutman, I., Polansky, O.E.: Mathematical Concepts in Organic Chemistry. Springer, Berlin (1986) · Zbl 0657.92024 [10] Gutman, I, Selected properties of the schultz molecular topological index, J. Chem. Inf. Comput. Sci., 34, 1087-1089, (1994) [11] Hou, YP; Li, JS, Bounds on the largest eigenvalues of trees with a given size of matching, Linear Algebra Appl., 342, 203-217, (2002) · Zbl 0992.05055 [12] Ilić, A; Stevanović, D; Feng, LH; Yu, GH; Dankelmann, P, Degree distance of unicyclic and bicyclic graphs, Discrete Appl. Math., 159, 779-788, (2011) · Zbl 1222.05040 [13] Ilić, A; Klavžar, S; Stevanović, D, Calculating the degree distance of partial Hamming graphs, MATCH Commun. Math. Comput. Chem., 63, 411-424, (2010) · Zbl 1265.05186 [14] B. McKay, Nauty. http://cs.anu.edu.au/bdm/nauty/ [15] Todeschini, R., Consonni, V.: Handbook of Molecular Descriptors. Wiley-VCH, Weinheim (2000) [16] Tomescu, I, Some extremal properties of the degree distance of a graph, Discrete Appl. Math., 98, 159-163, (1999) · Zbl 0936.05038 [17] Tomescu, I, Properties of connected graphs having minimum degree distance, Discrete Math., 309, 2745-2748, (2009) · Zbl 1182.05076 [18] Tomescu, I, Unicyclic and bicyclic graphs having minimum degree distance, Discrete Appl. Math., 156, 125-130, (2008) · Zbl 1132.05018 [19] Tomescu, I, Ordering connected graphs having small degree distances, Discrete Appl. Math., 158, 1714-1717, (2010) · Zbl 1208.05021 [20] Tomescu, I; Kanwal, S, Ordering connected graphs having small degree distances. II, MATCH Commun. Math. Comput. Chem., 67, 425-437, (2012) · Zbl 1289.05147 [21] Wiener, H, Structural determination of paraffin boiling point, J. Amer. Chem. Soc., 69, 17-20, (1947) [22] Yu, A; Tian, F, On the spectral radius of unicyclic graphs, MATCH Commun. Math. Comput. Chem., 51, 97-109, (2004) · Zbl 1053.05085
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