Let G be a graph with connected components, give adjacency matrix.

Question

Let G be a graph with connected components $G_1,\: \ldots ,\: G_l$, and $V(G_1)=\{u_1^1,\: u_2^1,\: \ldots ,\: u_{n_1}^1\},\: V(G_l)=\{u_1^l,\: u_2^l,\: \ldots ,\: u_{n_l}^l\}$ their set of vertices, respectively. Give the shape of the adjacency matrix of $G$.

Answer 1

write $V(G_i)=\{u_1^i,...,u_{n_i}^i\}$ where $|V(G_i)|=n_i$. Becuase $V=\bigcup_{i=1}^{n} V(G_i)$ and becuase the $G_i$-s are different connected components of $G$ whe have $[Adj]_{u,v} = 0$ whenever $u$ and $v$ are in diffrent components. Now let $A_i$ be the adjecency matrix of $G_i$, now we can give a full description of $Adj_G$ if $u,v$ are in diffrent components of $G$ we have $[Adj_{u,v}] = 0$ and if they are in the same component then $[Adj]_{u,v} = [A_i]_{u,v}$ so:

$$Adj = diag(A_1,A_2,...,A_l)$$