What is Graph, adjacency matrix?

A graph consists of vertices $1, 2, \dots, n$ and edges joining certain pairs of vertices (ordered pairs for a directed graph). Its adjacency matrix is the $n \times n$ matrix $M$ with $m_{ij} = 1$ if there is an edge from $i$ to $j$, and $0$ otherwise. A walk of length $k$ from $i$ to $j$ is a sequence of $k$ consecutive edges leading from $i$ to $j$.