Abstract
Let G be a connected graph with vertex set V(G). The degree distance of G is defined as \({D'(G) = \sum_{\{u, v\}\subseteq V(G)} (d_G(u) + d_G (v))\, d(u,v)}\), where d G (u) is the degree of vertex u, d(u, v) denotes the distance between u and v, and the summation goes over all pairs of vertices in G. In this paper, we characterize n-vertex unicyclic graphs with given matching number and minimal degree distance.
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References
Bondy J.A., Murty U.S.R.: Graph Theory with Applications. Macmillan Press, New York (1976)
Bucicovschi O., Cioabă S.M.: The minimum degree distance of graphs of given order and size. Discret. Appl. Math. 156, 3518–3521 (2008)
Chang A., Tian F.: On the spectral radius of unicyclic graphs with perfect matching. Linear Algebra Appl. 370, 237–250 (2003)
Dankelmann P., Gutman I., Mukwembi S., Swart H.C.: On the degree distance of a graph. Discret. Appl. Math. 157, 2773–2777 (2009)
Diudea M.V., Gutman I., Jäntschi L.: Molecular Topology. Nova, Huntington (2001)
Dobrynin A., Entringer R., Gutman I.: Wiener index of trees: theory and applications. Acta Appl. Math. 66, 211–249 (2001)
Dobrynin A.A., Kochetova A.A.: Degree distance of a graph: a degree analogue of the Wiener index. J. Chem. Inf. Comput. Sci. 34, 1082–1086 (1994)
Du Z., Zhou B.: Minimum Wiener indices of trees and unicyclic graphs of given matching number. MATCH Commun. Math. Comput. Chem. 63, 101–112 (2010)
Gutman I.: Selected properties of the Schultz molecular topological index. J. Chem. Inf. Comput. Sci. 34, 1087–1089 (1994)
Gutman I., Polansky O.E.: Mathematical Concepts in Organic Chemistry. Springer, Berlin (1986)
Hou Y.P., Li J.S.: Bounds on the largest eigenvalues of trees with a given size of matching. Linear Algebra Appl. 342, 203–217 (2002)
Ilić A., Stevanović D., Feng L., Yu G., Dankelmann P.: Degree distance of unicyclic and bicyclic graphs. Discret. Appl. Math. 159, 779–788 (2011)
Ilić A., Klavžar S., Stevanović D.: Calculating the degree distance of partial Hamming graphs. MATCH Commun. Math. Comput. Chem. 63, 411–424 (2010)
Ilić A.: Trees with minimal Laplacian coefficients. Comp. Math. Appl. 59, 2776–2783 (2010)
McKay, B.: Nauty http://cs.anu.edu.au/~bdm/nauty/
Tomescu I.: Some extremal properties of the degree distance of a graph. Discret. Appl. Math. 98, 159–163 (1999)
Tomescu I.: Properties of connected graphs having minimum degree distance. Discret. Math. 309, 2745–2748 (2009)
Tomescu A.I.: Unicyclic and bicyclic graphs having minimum degree distance. Discret. Appl. Math. 156, 125–130 (2008)
Tomescu, A.I.: Minimal graphs with respect to the degree distance, University of Bucharest. Technical report (2008). http://sole.dimi.uniud.it/~alexandru.tomescu/files/dd-distance.pdf
Yu A., Tian F.: On the spectral radius of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 51, 97–109 (2004)
Yuan H., An C.: The unicyclic graphs with maximum degree distance. J. Math. Study 39, 18–24 (2006)
Zhou B.: Bounds for the Schultz molecular topological index. MATCH Commun. Math. Comput. Chem. 56, 189–194 (2006)
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Feng, L., Liu, W., Ilić, A. et al. Degree Distance of Unicyclic Graphs with Given Matching Number. Graphs and Combinatorics 29, 449–462 (2013). https://doi.org/10.1007/s00373-012-1143-5
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DOI: https://doi.org/10.1007/s00373-012-1143-5