Adjacency Matrix Of A Graph In Pdf Form

4. Null Spaces of the Adjacency Matrix We begin with the two null spaces NA G and NAT G these two are the easiest to interpret in the context of graphs. At the end of each calculation, I will place a moral which explains precisely the connection between a fundamental subspace of the adjacency matrix and its interpretation in the world of

Adjacency Matrix An easy way to store connectivity information - Checking if two nodes are directly connected O1 time Make an n n matrix A - aij 1 if there is an edge from i to j - aij 0 otherwise Uses n2 memory - Only use when n is less than a few thousands, - and when the graph is dense Adjacency Matrix and Adjacency List 7

Adjacency quotlistquot conceptual representation Vertices Set of vertex labels SetltIntegergt for example graph Adjacencies Dictionary mapping from vertex labels to sets of vertex labels MapltInteger, SetltIntegergtgt for example graph But the reason for the name quotlistquot is you can do it all with just linked lists 1 2 3 4

3.2 Reachability in Graphs Denote element , ij of a matrix M by M ij. Adjacency matrix A shows which nodes can be reached from each other in one step, for aifvvEij j i 0,. Matrix A2 shows which nodes can be reached from each other in two steps, for 02 A ij means there are edges , ,, vv E v v Ejk ki form some node k. Generally

The adjacency matrix of an edgeless graph is a zero matrix. Complete graph The adjacency matrix of a complete graph is such that all entries are 1 except for the main diagonal entries which are all 0 Connected components Let G be a graph with connected components G 1, G 2, , G k. If each connected component G i has n i vertices numbered

Proof sketch. Given isomorphic graphs, the isomorphism gives a permu-tation of the vertices, which leads to a permutation matrix. Similarly, the permutation matrix gives an isomorphism. Now we see that the adjacency matrix can be used to count uv-walks. Theorem 3 Let A be the adjacency matrix of a graph G, where V G fv 1v 2v

a small graph, as the number of vertices and edges grows, it becomes harder to keep track of all the different ways the vertices are connected. Matrix notation and computation can help to answer these questions. The adjacency matrix for a graph with n vertices is an nn matrix whose i,j entry is 1

Block Diagonal Adjacency Matrix The nodes in a graph with pcomponents can be numbered so that the adjacency matrix has a block diagonal form with pblocks. That is, Ais a matrix with smaller square matrices along the main diagonal, and off-diagonal elements of 0.

42 Basic Concepts of Graphs 1.10 Matrix Representation of Graphs Denitions In this section, we introduce two kinds of matrix representations of a graph, that is, the adjacency matrix and incidence matrix of the graph. A graph Gwith the vertex-set VG x1,x2,,vv can be described by means of matrices. The adjacency matrix of Gis a v

If the graph is directed, we may still dene a signed adjacency matrix A with elements A ij 8 gtlt gt 1, if edge goes from v i to v j 1, if edge goes from v j to i 0, otherwise 1.6 The characteristic polynomial of a graph is dened as the characteristic polynomial of the adjacency matrix pG x detA xI1.7 For the graph in Fig. 1