How to define a transition matrix mathematically?

Question

I'm writing my master thesis. Given the adjacency matrix of a graph, I need to define the transition matrix formally. I'm not able to figure out how to define it in mathematical notation. Can you help please? The formal mathematical definition must say that each row of the this matrix is just the corresponding row in the adjacency matrix divided by the degree of the node that corresponds to the row.

Answer 1

Given an undirected graph $G$ with $n$ vertices $v_1, \ldots, v_n$, let $a_{i,j}$, $1\le i,j\le n$, be the number of edges from $v_i$ to $v_j$ (so that $a_{i,j}\in\{0,1\}$ if $G$ is a simple graphs). Then $\rho(v_i):=\sum_{j=1}^n a_{i,j}$ is called the degree of vertex $v_i$. The matrix $$A=(a_{i,j})_{i,j=1}^n$$ is called the adjacency matrix and, assuming that $\rho(v_i)>0$ for all vertices, $$T=(\tfrac1{\rho(v_i)}a_{i,j})_{i,j=1}^n$$ is called the transition matrix of the graph $G$.

Given $A$, we can compute $T$ without explicit regress to propeties of $G$ by noting that there is a unique diagonal matrix $D$ such that $$ A j= Dj$$ where $j=\begin{pmatrix}1\\\vdots\\1\end{pmatrix}$. (In fact, the diagonal entries od $D$ are the vertex degrees). Then $T=D^{-1}A$.