Degree distance of unicyclic and bicyclic graphs

  • Authors:

  • Aleksandar Ilić;Dragan Stevanović;Lihua Feng;Guihai Yu;Peter Dankelmann

  • Affiliations:

  • Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia;University of Primorska-FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia and Mathematical Institute, Serbian Academy of Science and Arts, Knez Mihajlova 36, 11000, Belgrade, Serbia;Department of Mathematics, Central South University, Railway Campus, Changsha, Hunan, 410075, PR China and School of Mathematics, Shandong Institute of Business and Technology, 191 Binhaizhong Roa ...;School of Mathematics, Shandong Institute of Business and Technology, 191 Binhaizhong Road, Yantai, Shandong, 264005, PR China;School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4000, South Africa

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2011

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Abstract

Let G be a connected graph with vertex set V(G). The degree distance of G is defined as D^'(G)=@?"{"u","v"}"@?"V"("G")(deg"G(u)+deg"G(v))d(u,v), where deg"G(u) is the degree of vertex u, and d(u,v) denotes the distance between u and v. Here we characterize n-vertex unicyclic graphs with girth k, having minimum and maximum degree distance, respectively. Furthermore, we prove that the graph B"n, obtained from two triangles linked by a path, is the unique graph having the maximum degree distance among bicyclic graphs, which resolves a recent conjecture of Tomescu.