# Thread: Graph Theory - Size of a Line Graph

1. ## Graph Theory - Size of a Line Graph

Hi, all!

Line graph $\displaystyle L(G)$ of a graph G is a graph in which every edge in $\displaystyle E(G)$ is represented with a vertex. Two vertices in $\displaystyle L(G)$ are adjacent if and only if the corresponding edges in G share a vertex.

Now suppose graph G has $\displaystyle n$ vertices, labeled $\displaystyle v_1, v_2, \dots, v_n$ and the degree of each vertex is $\displaystyle deg(v_i) = r_i$.

Find the size of $\displaystyle L(G)$.

I have attempted to solve it, but I'm stuck.

The order of $\displaystyle L(G)$ is $\displaystyle \frac{1}{2} \sum^{n}_{i=1} r_i$. Let's call it m.
The vertex $\displaystyle v \in V(L(G))$ is the edge from some vertex $\displaystyle u_i \in V(G)$ to another $\displaystyle u_k \in V(G)$, it's degree is therefore $\displaystyle deg(v_j) = (deg(u_{i_j}) - 1) + (deg(u_{k_j}) - 1) = r_{i_j} + r_{k_j} - 2$.
Size of $\displaystyle L(G) = \frac{1}{2} \sum^{m}_{j=1}(r_{i_j}+r_{k_j}) - m$

While I'm quite sure this is correct, it doesn't seem to be very useful - or indeed the expected answer. Any help?

2. Okay, I think I got now. If you write out the terms, you see that each $\displaystyle r_i$ is repeated exactly $\displaystyle r_i$ times. We can then write the size of $\displaystyle L(G)$ as $\displaystyle |E(L(G))| = \frac{1}{2}\sum^{n}_{i=1}{r_i^2} - m$.