A tree T , in an edge-colored graph G, is called a rainbow tree if no two edges of T are assigned the same color. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S ⊆ V (T ). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rxk(G). G...

The generalized prism πG of G is the graph consisting of two copies of G, with edges between the copies determined by a permutation π acting on the vertices of G. We define a generalized Cartesian product G π H that corresponds to the Cartesian product G H when π is the identity, and the generalized prism when H is the graph K2. Burger, Mynhardt and Weakley [On the domination number of prisms o...

In this paper, we provide exact value of the multiplicative version of the reciprocal degree distance and the multiplicative version of the reciprocal Gutman index of Cartesian product of complete graphs. Also, we establish sharp upper bounds for the multiplicative version of the reciprocal degree distance and multiplicative version of the reciprocal Gutman index of strong product of graphs.

The energy of a simple graph G is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two graphs of the same order are said to be equienergetic if they have the same energy. Several ways to construct equienergetic non-cospectral graphs of very large size can be found in the literature. The aim of this work is to construct equienergetic non-cospectral graphs of small size....

In this paper, we design the first linear-time algorithm for computing the prime decomposition of a digraph G with regard to the cartesian product. A remarkable feature of our solution is that it computes the decomposition of G from the decomposition of its underlying undirected graph, for which there exists a linear-time algorithm. First, this allows our algorithm to remain conceptually very s...

We enumerate prime graphs with respect to the Cartesian multiplication of graphs by expressing the species of connected graphs as the exponential composition of the species of prime graphs, where the exponential composition of species is constructed based on the arithmetic product of species of Maia and Méndez and the quotient species.

for two normal edge-transitive cayley graphs on groups h and k which have no common direct factor and gcd(jh=h ′j; jz(k)j) = 1 = gcd(jk=k ′j; jz(h)j), we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.