I am quite new to Algebraic Graph Theory. Can anyone give me a proof of the following fact:
If $G$ is a simple undirected graph and $H$ is a subgraph of $G$, then the largest eigenvalue of $H$ is at most the largest eigenvalue of $G$.
There is a very simple way to prove "If $A$ is a symmetric matrix and $B$ is the matrix obtained by deleting the last row and column of $A$, then the largest eigenvalue of $B$ is at most that of $A$" $-$ however, this corresponds only to the case when $H$ is obtained by deleting vertices from $G$.
Any help appreciated!