Adjacency and Incidence Matrices
1 / 10 The Incidence Matrix of a Graph Definition Let G = (V , E) be a graph where V = {1, 2,..., n} and E = {e1, e2,..., em}. 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 1 1 1 f 1 0 0 3 B = g 0 1 0 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 1 1 1 f 1 0 0 3 B = g 0 1 0 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 1 1 1 0 f h 1 0 0 1 3 B = g 0 1 0 1 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 = {1, 2,..., n}. 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 0 1 1 1 f 1 0 0 0 3 A = g 1 0 0 0 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.
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