The Cartesian product of two hamiltonian graphs is always hamiltonian. For directed graphs, the analogous statement is false. We show that the Cartesian product C,,, x C,, of directed cycles is hamiltonian if and only if the greatest common divisor (g.c.d.) d of n, and n, is a t least two and there exist positive integers d,, d, so that d, + d, = d and g.c.d. (n,, d,) = g.c.d. (n,, d,) = 1. We ...

Suppose ρ is a simple graph, then its eccentric harmonic index defined as the sum of terms id="M2"> 2 / e a + b for edges id="M3"> v , where id="M4"> eccentricity id="M5"> th vertex graph id="M6"> . ...

This article attempts to discuss the problems in building new topologies utilizing embedding. We propose to utilize the cartesian product to describe graphs as a right mathematical solution to adjacency matrix. We applied the cartesian product of two and sets of multiple dimensions in analysis. We proposed methodology of building logical topologies based on utilization of regular and symmetric ...

We examine the distinguishing number of the Cartesian product of an arbitrary number of complete graphs. We show that for u1 ≤ · · · ≤ ud the distinguishing number of the Cartesian product of complete graphs of these sizes is either du d e or du 1/s d e + 1 where s = Πd−1 i=1 ui. In most cases, which of these values it is can be explicitly determined.

We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.