# Simple graphs and adjacency matrices

Probability Level 2

Let $G$ be a simple graph with $n$ vertices and $m$ edges: that is, $G$ is undirected, unweighted, and has no loops (edges from a vertex to itself).

Let $A$ be the adjacency matrix of $G$ and let ${\bf u} = \begin{pmatrix} 1&1&\ldots&1 \end{pmatrix}$ be the $1 \times n$ matrix whose entries are all equal to $1.$ Then ${\bf u} A {\bf u}^T$ is a $1 \times 1$ matrix $\begin{pmatrix} b \end{pmatrix}$. What is $b$?

Notation: if $B$ is a matrix, $B^T$ denotes its transpose.

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