This mathematical problem is based on the assigned Barabasi Ch$.2$ reading. Newman Ch$.6$ is also

relevant. It does require prior understanding of matrix math, in particular multiplication and the trace

operator.

$($a$)$ Construct $($in a table below$)$ the matrix representation of the undirected graph pictured. The vertex

labels should index to the rows and columns of the matrix. You will use this matrix in subsequent

problems.

For the following sub$-$problems, let $n=|V|$ be the number of vertices, and let $A$ be the $nn$ adjacency

matrix of an arbitrary undirected network $($such as what you wrote above, but your responses

should work for any A$).$ Also let $1$ be the $n1$ column vector whose elements are all $1.$ Answer the

following using only matrix operations such as multiplication by another matrix or a constant, taking

the transpose, or taking the trace. DO NOT USE SUMMATION NOTATION $($which is implicit iteration$)$;

DO NOT WRITE LOOPS: the point here is to show how the metrics correspond to matrix math, not to

invoke linear procedures such as summation or iteration. If you get the wrong answer working out

your example, consider whether you are counting things multiple times, and adjust your expression.

$($b$)$ Write a general expression for the vector $k$ whose elements are the degrees of the vertices. Your

expression should work for any matrix representing a graph. Explain why this expression is correct.

Then illustrate how your expression works by carrying out the math with your matrix from part $($a$).$

$($c$)$ Write a general expression for the number $m=|E|$ of edges in the network. Again, the expression

should work for any graph. Explain why this expression is correct. Then illustrate how your expression

works by carrying out the math with your matrix from part $($a$).$

$($d$)$ Write an expression for $N$ where $N[i,j]$ is the number of common neighbors of vertices i and $j.$

Explain why this expression is correct. Then illustrate how your expression works by carrying out the

math with your matrix from part $($a$).$

$($e$)$ Write an expression for the total number of triangles in the network, where a triangle means three

vertices, each connected by edges to both of the others. $($Hint: use trace here.$)$ Explain why this

expression is correct. Then illustrate how your expression works by carrying out the math with your

matrix from part $($a$).$