Problem B$.2($Graph Theory$).$ For this question, it is best if you perform your own

experiments and see what is happening. Resist the temptation to google what is known.

I want you to do lots of computations and make conjectures. Proofs are not required,

but if some observations are trivially provable by quoting some theorem then do so$.$ It's

a test of your mathematical creativity. Enjoy!

A graph $G$ consists of a pair of sets $(V,E),$ where $V($the set of vertices$)$ and $E($the

set of edges$)$ such that each einE is a $2-$subset $\{v_1,v_2\}$ of $V.$ If an edge $e=\{v_1,v_2\},$ we

say $e$ is an edge between the vertices $v_1$ and $v_2.$ This is also called an undirected, simple,

graph. The word simple refers to no loops $($i$.$e$.,$ vertices joined to itself$)$ or parallel

edges $($i$.$e$.,$ two edges between a pair of vertices$).$ We can represent a graph by using the

adjacency matrix. The rows and columns are indexed by the vertices and we place a

$1$ if there is an edge between $v_i$ and $v_j.$ Here is an example of a graph on $5$ vertices.