The purpose of this paper is to present some coupled fixed point results on a metric space endowed with two $b$-metrics. We shall apply a fixed point theorem for an appropriate operator on the Cartesian product of the given spaces endowed with directed graphs. Data dependence, well-posedness and Ulam-Hyers stability are also studied. The results obtained here will be applied to prove the existe...

The distance $d(u,v)$ between two vertices $u$ and $v$ of a graph $G$ is equal to the length of a shortest path that connects $u$ and $v$. Define $WW(G,x) = 1/2sum_{{ a,b } subseteq V(G)}x^{d(a,b) + d^2(a,b)}$, where $d(G)$ is the greatest distance between any two vertices. In this paper the hyper-Wiener polynomials of the Cartesian product, composition, join and disjunction of graphs are compu...

We find exact values for the game chromatic number of the Cartesian product graphs Sm¤Pn, Sm¤Cn, P2¤Wn, and P2¤Km,n. This extends previous results of Bartnicki et al. on the game chromatic number of Cartesian product graphs.

Given a Cartesian product G = GI x x G,, (m > 2) of nontrivial connected graphs G, and the base d, dimension D de Bruijn graph B(d,D), it is investigated under which conditions G is (or is not) a subgraph of B(d, D). We present a complete solution of this problem for the case D 3 4. For D = 3, we give partial results including a complete solution for the case that G is a torus, i.e., G is the C...