doubt about basic definition I have a very basic doubt regarding my last post understanding the basic definiton
Can anybody clear the point what would be the result if I take  $j=1$. 
For ex. if I am taking $(a,x,1)$ and $(b,x,1)$ as the vertices, Will they be adjacent or not?
Here the value of $j$ is 1. I made them adjacent as acc to definition...
vertices $(x_1, x_2, . . . , x_k)$, $(y_1, y_2, . . . , y_k)$ are adjacent if for some index j $\in$  {1, 2, . . . , k} we
have $x_jy_j$ $\in$$E(G_j )$ and $x_i = y_i$ for each 1$\leq$ i < j.
In example I took $V(P_3)$ = {a,b,c,d}, $V(K_2)$= {x,y} and $V(K_2)$={1,2}.
And the product is $P_3$$\circ$$K_2$$\circ$$K_2$
Thanks a lot.
 A: If $x_1$ and $y_1$ are adjacent in $G_1$, then $\langle x_1,x_2,\dots,x_k\rangle$ and $\langle y_1,y_2,\dots,y_k\rangle$ are adjacent in $G_1\circ G_2\circ\ldots\circ G_k$ no matter what $x_2,\dots,x_k$ and $y_2,\dots,y_k$ are. In this case the condition that $x_i=y_i$ for $1<i\le j$ is vacuously satisfied, because there are no such $i$. Thus, in your example $\langle a,u,m\rangle$ is adjacent to $\langle b,v,n\rangle$ in $P_3\circ K_2\circ K_2$ for any choice of $u,v\in\{x,y\}$ and $m,n\in\{1,2\}$.
Here’s another way to describe the adjacency relation in $G_1\circ G_2\circ\ldots\circ G_k$. Given distinct vertices $x=\langle x_1,x_2,\dots,x_k\rangle$ and $y=\langle y_1,y_2,\dots,y_k\rangle$, let 
$$d(x,y)=\min\Big\{i\in\{1,\dots,k\}:x_i\ne y_i\Big\}\;;$$
then $x$ and $y$ are adjacent in $G_1\circ G_2\circ\ldots\circ G_k$ if and only if $x_{d(x,y)}$ and $y_{d(x,y)}$ are adjacent in $G_{d(x,y)}$. 
In words, examine the coordinates of $x$ and $y$ from left to right. Since $x\ne y$, you will eventually come to one at which $x$ and $y$ differ. Say that this first occurs at position $j$: then $x$ and $y$ are adjacent in $G_1\circ G_2\circ\ldots\circ G_k$ if and only if and only if $x_j$ and $y_j$ are adjacent in $G_j$. (This $j$ is the $d(x,y)$ of the previous paragraph.) This $j$ might be $1$, in which case you don’t even have to look at any other coordinate: if $x_1$ and $y_1$ are adjacent in $G_1$, then $x$ and $y$ are adjacent in $G_1\circ G_2\circ\ldots\circ G_k$. Or $x_1$ and $y_1$ might be equal, in which case you go on and look at $x_2$ and $y_2$. If $x_2\ne y_2$, then $x$ and $y$ are adjacent in $G_1\circ G_2\circ\ldots\circ G_k$ if and only if $x_2$ and $y_2$ are adjacent in $G_2$; if $x_2=y_2$, you go on and look at $x_3$ and $y_3$. And so on.
A: Yes, $(a,x,1)$ and $(b,x,1)$ will be adjacent, assuming that $ab \in E(P_3)$.
It might help to understand what part of this definition makes it lexicographic. As an analogy, consider alphabetizing words in a dictionary:

Which of the following words comes first in the dictionary: "$\text{snowman}$" or "$\text{snowball}$"?

To figure this out, we scan the letters of each word from left to right until we find two letters that don't match. The first four letters are the same, so we look at each word's fifth letter. Since '$\text{b}$' comes before '$\text{m}$' in the alphabet, "$\text{snowball}$" must come before "$\text{snowman}$" in the dictionary. Notice that the remaining letters to the right of these unmatched letters are ignored. Likewise, we can alphabetize the words "$\text{stack}$" and "$\text{exchange}$" by only looking at their first letters (regardless of the remaining letters).
Similarly, to figure out if two vectors of vertices are adjacent, scan along from left to right until you find two vertices that don't match. If these two unmatched vertices are connected by an edge, then the original two vectors of vertices are considered adjacent (regardless of the remaining vertices to the right).
