Adjacency and Incidence Matrices 1 / 10 The Incidence Matrix of a Graph Definition Let G = (V ; E) be a graph where V = f1; 2;:::; ng and E = fe1; e2;:::; emg. The incidence matrix of G is an n × m matrix B = (bik ), where each row corresponds to a vertex and each column corresponds to an edge such that if ek is an edge between i and j, then all elements of column k are 0 except bik = bjk = 1. 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 2 / 10 The First Theorem of Graph Theory Theorem If G is a multigraph with no loops and m edges, the sum of the degrees of all the vertices of G is 2m. Corollary The number of odd vertices in a loopless multigraph is even. 3 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 4 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 1 03 f h 61 0 0 17 3 B = 6 7 g 40 1 0 15 4 0 0 1 0 5 / 10 The Adjacency Matrix of a Graph Definition Let G = (V ; E) be a graph with no multiple edges where V = f1; 2;:::; ng. The adjacency matrix of G is the n × n matrix A = (aij ), where aij = 1 if there is an edge between vertex i and vertex j and aij = 0 otherwise. Notes The adjacency matrix of a graph is symmetric. 6 / 10 Adjacency Matrix Example 1 2 e 20 1 1 13 f 61 0 0 07 3 A = 6 7 g 41 0 0 05 4 1 0 0 0 7 / 10 Vertex Degree Definitions The degree of a vertex in a graph is the number of edges incident on that vertex. A vertex is odd if its degree is odd; otherwise, it is even. Notes The sum of the elements of row i of the adjacency matrix of a graph is the degree of vertex i. The sum of the elements of column i of the adjaceny matrix of a graph is the degree of vertex i. 8 / 10 Linear Algebra and Adjacency Matrices of Graphs Proposition Let A be the adjacency matrix of a graph. The (i; i)-entry in A2 is the degree of vertex i. Recall that the trace of a square matrix is the sum of its diagonal entries. Proposition Let G be a graph with e edges and t triangles. If A is the adjacency matrix of G, then (a) trace(A) = 0, (b) trace(A2) = 2e, (c) trace(A3) = 6t. 9 / 10 Acknowledgements Statements of definitions follow the notation and wording of Balakrishnan's Introductory Discrete Mathematics. 10 / 10.