On the Wiener Complexity and the Wiener Index of Fullerene Graphs

On Transmission Irregular Cubic Graphs of an Arbitrary Order

The transmission of a vertex v of a graph G is the sum of distances from v to all the other vertices of G. A transmission irregular graph (TI graph) has mutually distinct vertex transmissions. In 2018, Alizadeh and Klavžar posed the following question: do there exist infinite families of regular TI graphs? An infinite family of TI cubic graphs of order 118+72k, k≥0, was constructed by Dobrynin in 2019. In this paper, we study the problem of finding TI cubic graphs for an arbitrary number of vertices. It is shown that there exists a TI cubic graph of an arbitrary even order n≥22. Almost all constructed graphs are contained in twelve infinite families.

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 Djoković–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.

On correlation of hyperbolic volumes of fullerenes with their properties

We observe that fullerene graphs are one-skeletons of polyhedra, which can be realized with all dihedral angles equal to π /2 in a hyperbolic 3-dimensional space. One of the most important invariants of such a polyhedron is its volume. We are referring this volume as a hyperbolic volume of a fullerene. It is known that some topological indices of graphs of chemical compounds serve as strong descriptors and correlate with chemical properties. We demonstrate that hyperbolic volume of fullerenes correlates with few important topological indices and so, hyperbolic volume can serve as a chemical descriptor too. The correlation between hyperbolic volume of fullerene and its Wiener index suggested few conjectures on volumes of hyperbolic polyhedra. These conjectures are confirmed for the initial list of fullerenes.