Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

If G is a graph and P is a partition of V(G), then the partition distance of G is the sum of the distances between all pairs of vertices that lie in the same part of P. A colored distance is the dual concept of the partition distance. These notions are motivated by a problem in the facility location network and applied to several well-known distance-based graph invariants. In this paper, we apply an extended cut method to induce the partition and color distances to some subsets of vertices which are not necessary a partition of V(G). Then, we define a two-dimensional weighted graph and an operator to prove that the induced partition and colored distances of a graph can be obtained from the weighted Wiener index of a two-dimensional weighted quotient graph induced by the transitive closure of the Djokovic–Winkler relation as well as by any partition that is coarser. Finally, we utilize our main results to find some upper bounds for the modified Wiener index and the number of orbits of partial cube graphs under the action of automorphism group of graphs.

[1]  D. Djoković.  Distance-preserving subgraphs of hypercubes , 1973 .

[2]  N. Tratnik.  The Graovac–Pisanski index of zig-zag tubulenes and the generalized cut method , 2017, Journal of Mathematical Chemistry.

[3]  Sudeep Stephen,et al.  Average Distance in Interconnection Networks via Reduction Theorems for Vertex-Weighted Graphs , 2016, Comput. J..

[4]  Peter Winkler,et al.  Isometric embedding in products of complete graphs , 1984, Discret. Appl. Math..

[5]  J. Hatzl.  Median problems on wheels and cactus graphs , 2007, Computing.

[6]  K. R. Udaya Kumar Reddy,et al.  A survey of the all-pairs shortest paths problem and its variants in graphs , 2016 .

[7]  Aleksandar Ilic,et al.  Generalizations of Wiener Polarity Index and Terminal Wiener Index , 2011, Graphs Comb..

[8]  P. Slater,et al.  Minimean Location of Different Facilities on a Line Network , 1981 .

[9]  Wayne Goddard,et al.  On the graphs with maximum distance or $k$-diameter , 2005 .

[10]  Sandi Klavžar,et al.  Partition distance in graphs , 2017, Journal of Mathematical Chemistry.

[11]  Bojan Mohar,et al.  Labeling of Benzenoid Systems which Reflects the Vertex-Distance Relations , 1995, J. Chem. Inf. Comput. Sci..

[12]  Sandi Klavzar,et al.  Wiener Number of Vertex-weighted Graphs and a Chemical Application , 1997, Discret. Appl. Math..

[13]  W. Kruijt,et al.  Probabilistic analysis of the distance between clusters randomly distributed on the electrode surface , 1993 .

[14]  Sandi Klavzar,et al.  Wiener index in weighted graphs via unification of Θ∗Θ∗-classes , 2014, Eur. J. Comb..

[15]  Ante Graovac,et al.  On the Wiener index of a graph , 1991 .

[16]  Sandi Klavzar,et al.  Computing distance moments on graphs with transitive Djoković-Winkler relation , 2014, Discret. Appl. Math..

[17]  Sandi Klavzar,et al.  Wiener index versus Szeged index in networks , 2013, Discret. Appl. Math..

[18]  A. Dobrynin,et al.  On the Wiener Complexity and the Wiener Index of Fullerene Graphs , 2019, Mathematics.

[19]  Mihai V. Putz,et al.  Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems , 2020, Symmetry.

[20]  Matthias Dehmer,et al.  Topological mappings between graphs, trees and generalized trees , 2007, Appl. Math. Comput..

[21]  R. Graham,et al.  On isometric embeddings of graphs , 1985 .

[22]  Sandi Klavzar,et al.  Modified Wiener index via canonical metric representation, and some fullerene patches , 2016, Ars Math. Contemp..

[23]  Aleksandra Tepeh,et al.  Trees with the maximal value of Graovac-Pisanski index , 2019, Appl. Math. Comput..

[24]  Kinkar Chandra Das,et al.  On maximum Wiener index of trees and graphs with given radius , 2017, J. Comb. Optim..

[25]  Sandi Klavzar and Mohammad J. Nadjafi-Arani.  Cut Method: Update on Recent Developments and Equivalence of Independent Approaches , 2014 .

[26]  Danail Bonchev,et al.  Trends in information theory-based chemical structure codification , 2014, Molecular Diversity.