You can have the adjacency matrix of the line graph of the K5 regular graph by considering the edges as the columns and the rows. You will have to fill the matrix from the graph then, with an entry in a cell aij if edge ei and ej are adjacent to each other in the K5 graph. You will observe that every row (or column) will have exactly six entry, so we can say that the degree of the graph is 6. Also, since this is an undirected graph, so every vertex ( or the edge pair ) will be counted twice, one time for filling up the entry for edge ei and other time for filling up the entry for ej. Now, since every row (or column) have six entry and there are ten rows (or columns) and every actual vertex (of the K5 regular graph) is counted twice, we have : (6 * 10) / 2 = 30 edges.
