Graph Theory

Adjacency Matrices

$a_{ij}=1$ if there exists an edge from vertex $i$ to vertex $j$, otherwise $a_{ij}=0$, for an unweighted graph.

If the graph is weighted, $a_{ij}=w_{ij}$, where $w_{ij}$ represents the weight of the edge from $i$ to $j$, or $0$ if there exists no edge.

Transition Matrices

$a_{ij}$ is the probability of moving from vertex $i$ to vertex $j$ in a given step. (Note: Some curricula define $a_{ij}$ as the probability of moving from $j$ to $i$)