The purpose of this paper is to study the information ratio of perfect secret sharing of product of some special families of graphs. We seek to prove that the information ratio of prism graphs Yn are equal to 7/4 for any n³5, and we will gave a partial answer to a question of Csirmaz [10]. We will also study the information ratio of two other families Cm×Cn and Pm×Cn and obtain the exact value of information ratio of these graphs.

Reliability Wiener number is a modification of the original Wiener number in which probabilities are assigned to edges yielding a natural model in which there are some (or all) bonds in the molecule that are not static. Various probabilities naturally allow modelling different types of chemical bonds because chemical bonds are of different types and it is well known that under certain conditions the bonds can break with certain probability. This is fully taken into account in quantum chemistry. In the model considered here, probabilistic nature is taken into account and at the same time the conceptual simplicity of the discrete graph theoretical model is preserved. Here we extend previous studies by deriving a formula for the reliability Wiener number of a Cartesian product of graphs G-H.

The zero forcing number of a graph $G$, denoted $Z(G)$, is a graph parameter  which is based on a color change rule that describes how to color the vertices. Zero forcing is useful in several branches of science such as electrical engineering, computational complexity and quantum control.  In this paper, we investigate the zero forcing number for Cartesian products of some graphs. The main contribution of this paper is to introduce a new presentation of the Cartesian product of two complete bipartite graphs and to obtain the zero forcing number of these graphs.  We also introduce a purely graph theoretical method to prove $Z(K_n \Box K_m)=mn-m-n+2$.

Let $G$ be a simple connected graph with diameter $d$, and $k\in [1,d]$ be an integer. A radio $k$-coloring of graph $G$ is a mapping $g:V(G)\rightarrow \{0\}\cup \mathbb{N}$ satisfying $\lvert g(u)-g(v)\rvert\geq 1+k-d(u,v)$ for any pair of distinct vertices $u$ and $v$ of the graph $G$, where $d(u,v)$ denotes distance between vertices $u$ and $v$ in $G$. The number ${\text{max}} \{g(u):u\in V(G)\}$ is known as the span of $g$ and is denoted by $rc_k(g)$. The radio $k$-chromatic number of graph $G$, denoted by $rc_k(G)$, is defined as $\text{min} \{rc_k(g) : g \text{ is a radio $k$-coloring of $G$}\}$. For $k=d-1$, the radio $k$-coloring of graph $G$ is called an antipodal coloring. So $rc_{d-1}(G)$ is called the antipodal number of $G$ and is denoted by $ac(G)$. Here, we study antipodal coloring of the Cartesian product of the complete graph $K_r$ and cycle $C_s$, $K_r\square C_s$, for $r\geq 4$ and $s\geq 3$. We determine the antipodal number of $K_r\square C_s$, for even $r\geq 4$ with $s\equiv 1(mod\,4)$; and for any $r\geq 4$ with $s=4t+2$, $t$ odd. Also, for the remaining values of $r$ and $s$, we give lower and upper bounds for $ac(K_r\square C_s)$.

The chromatic number,  (G) of a graph G is the minimum number of colours used in a proper colouring of G. In an improper colouring, an edge uv is bad if the colours assigned to the end vertices of the edge is the same. Now, if the available colours are less than that of the chromatic number of graph G, then colouring the graph with the available colours lead to bad edges in G. The number of bad edges resulting from a  (k)-colouring of G is denoted by bk(G). In this paper, we use the concept of  (k)-colouring and determine the number of bad edges in Cartesian product of some graphs.

Let G be a graph and c′ aa(G) denotes the minimum number of colors required for an acyclic edge coloring of G in which no two adjacent vertices are incident to edges colored with the same set of colors. We prove a general bound for c′ aa (G □ H) for any two graphs G and H. We also determine exact value of this parameter for the Cartesian product of two paths, Cartesian product of a path and a cycle, Cartesian product of two trees, hypercubes. We show that c′ aa (Cm □ Cn) is at most 6 fo every m ≥ 3 and n≥3. Moreover in some cases we find the exact value of c′ aa (Cm □ Cn).

Let G be a graph. The first Zagreb polynomial M1 (G, x) and the third Zagreb polynomial M3 (G, x) of the graph G are defined as: M1 (G, x)= Se=uvÎE(G)x[d(u)+d(v)], M3 (G, x)= Se=uvÎE(G)x |d(u)-d(v)| . In this paper, we compute the first and third Zagreb polynomials of Cartesian product of two graphs and a type of dendrimers.

In this paper, we find a lower-bound for the information ratio of the Cartesian product of an arbitrary tree with diameter at least 3 and a cycle Cm for every m 3. Moreover, we determine the best information ratio of the perfect secret sharing scheme based on the graph constructed from the Cartesian product of a cycle of length 6 with the d-dimensional cube Qd. More precisely, it is shown that for every d 1, the information ratio of is exactly.

In this paper, we investigate a problem of finding natural condition to assure the product of two graphs to be hamilton-connected. We present some sucient and necessary conditions for GH being hamilton-connected when G is a hamilton-connected graph and H is a tree or G is a Hamiltonian graph and H is K2.