Gioan showed that the number of cycle reversing classes of totally cyclic orientations of a given graph can be calculated as an evaluation of the corresponding Tutte polynomial. We note that the concept of cycle reversing classes of orientations coincides with that of Eulerianequivalence classes considered by Chen and Stanley, and Kochol. Based on this coincidence, we give a bijective proof of ...

the wiener polarity index wp(g) of a molecular graph g of order n is the number ofunordered pairs of vertices u, v of g such that the distance d(u,v) between u and v is 3. in anearlier paper, some extremal properties of this graph invariant in the class of catacondensedhexagonal systems and fullerene graphs were investigated. in this paper, some new bounds forthis graph invariant are presented....

a multicone graph is defined to be join of a clique and a regular graph. a graph $ g $ is  cospectral with  graph $ h $ if their adjacency matrices have the same eigenvalues. a graph $ g $ is  said to be  determined by its spectrum or ds for short, if for any graph $ h $ with $ spec(g)=spec(h)$, we conclude that $ g $ is isomorphic to $ h $. in this paper, we present new classes of  multicone g...

The five problems of counting component colorings, vertex colorings, arc colorings, cocycles, and switching equivalence classes of a graph with respect to a finite field up to isomorphism are related by an exact sequence that stems from a coboundary operator. This cohomology is presented, and counting formulas are given for each of the five problems.

It has recently been established that there are exactly seven Witt equivalence classes of quadratic number fields, and then all quadratic and cubic number fields have been classified with respect to Witt equivalence. In this paper we have classified number fields of degree four. Using this classification, we have proved the Conjecture of Szymiczek about the representability of Witt equivalence ...