4
votes
1answer
166 views

Divisibility Relation for Spanning Trees of a Graph

Let $A = \big[{1\ 1\atop 1\ 0}\big]$, and let $G_n$ be the graph whose adjacency matrix is $A^{\otimes n}$. Also let $\kappa(G)$ denote the number of spanning trees of $G$. From a significant amount ...
6
votes
3answers
1k views

Eigenvectors and partitions of graphs

Let G be an undirected graph with the node set $V$ and the Laplacian matrix $L$. Let $N(v)$ denote the neighbors of a node $v$ and $|N(v)|$ its degree. Then a partition $\pi=(V_1, V_2, \ldots, V_k)$ ...