(Graph Theory) Adjacency Matrix_the adjacency matrix

This post will briefly go over the definition of adjacency matrix for graphs, some related theorems regarding graph isomorphism, and some proofs.

Adjacency Matrix

• The adjacency matrix is one way to formally represent a graph, for example in convenience for computer computation.
• Given a graph $G:=(V,E)$, where $|V|=n$, to obtain the adjacency matrix of this graph, we firstly select an ordering of the vertices, say we (re)name them as $v_1,...,v_n$.
• Next, we label the rows and columns of a matrix with the ordered vertices.
• The entry in this matrix in row $i4, column $j$, when $i\ne j$, is the number of edges incident on $i$ and $j$.
• If $i=j$, the entry is twice the number of loops incident on $i$, and this is because we want the nice formula ${\sum }_{i=1}^n\delta (v_i)=2|E|,n\ge 1$ to hold in general ($n=0$ is not particuarly interesting).

Remark

• Besides entries on the main diagonal, the information are redundantly stored twice in the adjacency matrix (the information above the main diagonal is identical to the information below the main diagonal). Therefore, to obtain $\delta (v_i)$, we sum up all the entries in the $i$th row OR all the entries in the $i$th column, but never both.

Reordering the Vertices

• Ordering affects which row represents which vertex. Observe that if a graph has $n$ vertices, we have $n!$ ways to reorder the vertices, and we can thus produce $n!$ (not necessarily distinct) adjacency matrices.

Graph Isomorphism and Adjacency Matrix

Proposition

• Two graphs $G_1,G_2$ are isomorphic iff there is some ordering of their vertices so that $A_1=A_2$ ($A_1$ is an adjacency matrix of $G_1$, and $A_2$ is an adjacency matrix of $G_2$).

Proof

Isomorphism$⟹$Ordering

• Suppose $G_1,G_2$ are isomorphic, then there exist bijections $f:V_1\to V_2$ and $g:E_1\to E_2$ such that $e\in E_1$ is incident on $v,w\in V_1$ iff $g(e)\in E_2$ is incident on $f(v),f(w)\in V_2$.
• Thus, we can choose arbitrarily an ordering of the vertices of $G_1$, say $v_1,...,v_n$, which gives us an adjacency matrix of $G_1$, $A_1$. Since $f$ is a bijection, we can let the ordering of vertices of $G_2$ be f ( v