Studying algebraic graph theory, I've stumbled across a wide range of results that I found pretty stunning and also useful. I'd like to share them here and ask for your favorite result from this area?

Let $G$  be a simple graph  $A$ and $L$ its adjacency and Laplacian matrix $\lambda_1\leq \cdots \leq\lambda_n$ and $\mu_1 \leq \cdots \leq \mu_n$ the respective eigenvalues of $A$ and $L$.

 1. (Wilf, Hoffman) For a nontrivial graph $G$  $$1+\frac{\lambda_n}{-\lambda_1} \leq \chi(G) \leq 1+\lambda_n$$
 

 2. (Sachs,Harary) Let $G$ be a graph of odd girth $2r+1.$ Let $p(x) = \sum_{i=0}^n c_{n-i} x^i$ be the characteristic polynomial of $A.$ Then $$c_3 = \cdots = c_{2r-1} = 0$$ and $\frac{-c_{2r+1}}{2}$ is the number of $(2r+1)$-cycles in G.



 3. (Folklore) The number of triangles in $G$ equals $tr(A^3)/6.$ Since matrix multiplication can be done in $O(n^k)$ for $k \leq 2.37$ this presents an improvement over the straightforward approach for counting triangles in graphs that is currently the fastest way to compute the number of triangles in simple graphs.

 4. (Kirchhoff) The number of spanning trees in G is $\frac{1}{n}\mu_2 \cdots \mu_n$

 5. (McKay) $diam(G) \geq  \frac{4}{n\mu_2}$


Do you have any neat results like this to share?