![]() ![]() If the graph is not connected, you can similarly tell the connected components from the rows of $A^k$. ![]() So if you have a row in $A^k$ that is all non-zero, then the graph is connected. If a graph has n number of vertices, then the adjacency matrix of that graph is n x n, and each entry of the matrix represents the number of edges from one vertex to another. It is the 2D matrix that is used to map the association between the graph nodes. there is a path of length 2 between i and j if k j and k i and there is a path of length 1. So if you start with $A$ and keep squaring until you get $A^k$ where $k \geq n$ where $n$ is the number of nodes, then the non-zero entries in row $i$ tell you all the nodes that are connected to node $i$ (since two connected nodes must be connected by a path of length $n$ or less). In graph theory, an adjacency matrix is a dense way of describing the finite graph structure. An adjacency matrix is defined as follows: Let G be a graph with n vertices that are assumed to be ordered from v1 to vn. If you put all 1 on the diagonal of your adjacency matrix A, and all edge weights are positive then when you multiply A 2 A A you get a non-zero entry a i j in A 2 if and only if there exist non-zero a i k and a k j in A for some k, i.e. Similarly the entries in $A^k$ tell you all pairs of nodes that are connected by a path of length $k$. So the non-zero entries in $A^2$ tell you all pairs of nodes that are connected by a path of length $2$. The matrix values correspond to the weights, or cost, of traversal to another vertex. For a graph of V vertices, the matrix representation has a dimension of V x V. Each node, or vertex, is represented as a column and row entry. It is a part of Class 12 Maths and can be defined as a matrix. A finite graph can be represented in the form of a square matrix on. The adjacency matrix is often also referred to as a connection matrix or a vertex matrix. there is a path of length $2$ between $i$ and $j$ if $k\neq j$ and $k\neq i$ and there is a path of length $1$ if $k = j$ or $k = i$. The first option, an adjacency matrix, is perhaps the most intuitive representation of our graph. An adjacency matrix is a way of representing a graph as a matrix of booleans (0s and 1s). If you put all 1 on the diagonal of your adjacency matrix $A$, and all edge weights are positive then when you multiply $A^2 = A*A$ you get a non-zero entry $a_$ in $A$ for some $k$, i.e. ![]()
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