# Intuitive interpretation of the adjacency matrix as a linear operator.

Naturally we can describe graphs via tables of "yes there is an edge" or "no there is not" between each pair of vertices, so the definition of an adjacency matrix is easily understood. Thinking of these tables as matrices, however, adds structure - specifically, an interpretation as a linear operator. Why do we look at them in this light? Is it just for application - for example, efficiently obtaining a lot of data about a graph by computing its spectrum? Or is there also an intuitive geometric (or algebraic) motivation behind the adjacency matrix?

For example, the $2$-path has adjacency matrix $$\mathcal{A}(P_2)=\left(\begin{array}{cc} 0 & 1\\1 & 0\end{array}\right)$$ which acts on a $2$-dimensional vector space by flipping the coordinates, $(x,y)\mapsto (y,x)$. Can we somehow intuitively connect this action to the $2$-path? What about for other simple graphs?

If $G = (V, E)$ is a graph, then the adjacency matrix is an operator which acts on the space of functions $V \to \mathbb{R}$ (say) via
$$A(f)(v) = \sum_{v \to w} f(w).$$