Simple graphs and adjacency matrices

Probability Level 2

Let GG be a simple graph with nn vertices and mm edges: that is, GG is undirected, unweighted, and has no loops (edges from a vertex to itself).

Let AA be the adjacency matrix of GG and let u=(111){\bf u} = \begin{pmatrix} 1&1&\ldots&1 \end{pmatrix} be the 1×n1 \times n matrix whose entries are all equal to 1.1. Then uAuT{\bf u} A {\bf u}^T is a 1×11 \times 1 matrix (b)\begin{pmatrix} b \end{pmatrix}. What is bb?

Notation: if BB is a matrix, BTB^T denotes its transpose.

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