Digraphs with large maximum Wiener index

Recently the concept of Wiener index was extended to digraphs which are not-necessarily strongly connected, and it was shown that some fundamental results extend naturally within this concept. This extension could be applicable in the topics of directed large networks, particularly because with this measure, one assigns finite values to the average distance and betweenness centrality of the nodes in a directed network. It is not hard to show that among digraphs on n vertices, the directed cycle C ? n achieves the maximum Wiener index. Next, we investigate digraphs with the second maximum Wiener index. One can consider this problem in the realm of all digraphs or restricted to those obtained by directing undirected graphs, so called antisymmetric digraphs. In both situations, we obtain that such digraphs are constructed from C ? n (n ? 6) by adding a single arc. We conclude the paper with consideration for possible further works.

[1]  Yongtang Shi,et al.  On Wiener polarity index of bicyclic networks , 2016, Scientific Reports.

[2]  Roger C. Entringer,et al.  Distance in graphs , 1976 .

[3]  Shujuan Cao,et al.  Degree-based entropies of networks revisited , 2015, Appl. Math. Comput..

[4]  Aleksandra Tepeh,et al.  Orientations of graphs with maximum Wiener index , 2016, Discret. Appl. Math..

[5]  D. Bonchev.  On the complexity of directed biological networks , 2003, SAR and QSAR in environmental research.

[6]  Kexiang Xu,et al.  A Survey on Graphs Extremal with Respect to Distance-Based Topological Indices , 2014 .

[7]  Danail Bonchev,et al.  Complexity of Protein-Protein Interaction Networks, Complexes, and Pathways , 2003 .

[8]  John W. Moon.  On the total distance between nodes in tournaments , 1996, Discret. Math..

[9]  Aleksandra Tepeh,et al.  Mathematical aspects of Wiener index , 2015, Ars Math. Contemp..

[10]  Aleksandra Tepeh,et al.  Some remarks on Wiener index of oriented graphs , 2016, Appl. Math. Comput..

[11]  Yongtang Shi,et al.  Graph distance measures based on topological indices revisited , 2015, Appl. Math. Comput..

[12]  Hong Lin.  On the Wiener Index of Trees with Given Number of Branching Vertices , 2014 .

[13]  Ortrud R. Oellermann,et al.  Minimum average distance of strong orientations of graphs , 2004, Discret. Appl. Math..

[14]  J. K. Doyle,et al.  Mean distance in a graph , 1977, Discret. Math..

[15]  I. Gutman,et al.  Wiener Index of Trees: Theory and Applications , 2001 .

[16]  Ján Plesník,et al.  On the sum of all distances in a graph or digraph , 1984, J. Graph Theory.

[17]  Aleksandra Tepeh,et al.  An inequality between the edge-Wiener index and the Wiener index of a graph , 2015, Appl. Math. Comput..

[18]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[19]  Ben Shneiderman,et al.  Structural analysis of hypertexts: identifying hierarchies and useful metrics , 1992, TOIS.

[20]  Matthias Dehmer,et al.  A history of graph entropy measures , 2011, Inf. Sci..

[21]  Xueliang Li,et al.  Novel inequalities for generalized graph entropies - Graph energies and topological indices , 2015, Appl. Math. Comput..

[22]  Frank Harary,et al.  Status and Contrastatus , 1959 .

[23]  S. Borgatti,et al.  Betweenness centrality measures for directed graphs , 1994 .

[24]  Aleksandra Tepeh,et al.  A Congruence Relation for the Wiener Index of Graphs with a Tree-Like Structure , 2014 .

[25]  Jing Ma,et al.  The Wiener Polarity Index of Graph Products , 2014, Ars Comb..

[26]  Ivan Gutman,et al.  Vertex Version of the Wiener Theorem , 2014 .

[27]  Martin Knor,et al.  Wiener Index of Line Graphs , 2014 .