A line graph L$($G$)$ of a given graph G $($V$,$E$)$ is created as follows; Each edge of G is a vertex in the

line graph. Two vertices in L$($G$),$ i$.$e$.$ edges in G$,$ are connected if they share a common vertex in G$.$

The incidence matrix B$($G$)$ of graph G is a matrix with $|$V$|$ rows and $|$E$|$ columns, where B$($i$,$j$)=1$ is

vertex i is incident on edge j and $0$ otherwise. Let A$($L$($G$))$ be the adjacency matrix of L$($G$).$ Given

this information answer the following questions $(40)$

$($a$)$ Prove or disprove that if two graphs are connected then their line graphs are also

connected. $(10)$

$($b$)$ Give an example where if two simple graphs are non$-$isomorphic, but the line graphs are

isomorphic. Draw the two graphs and their line graphs. There is a unique example, so you

might have to search the internet for the example. $(10)$

$($c$)$ Prove that A$($L$($G$)=$ B$($G$)$ T B$($G$)-2$I, where I is the incidence matrix. $(20)$