An adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal.
Questions tagged [adjacency-matrix]
253 questions
36
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Finding path-lengths by the power of Adjacency matrix of an undirected graph
I knew from Mark Newman's book - Networks: An Introduction (Page 137, Eq: 6.31) that, if $A$ is the adjacency matrix of a graph, then $ij$'th entry of $A^k$ will give me the number of $k$-length paths connecting the vertices $i$ and $j$. This…
kada mati
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On the invertibility of the adjacency matrix of a graph
Which are the sufficient and necessary conditions for an undirected graph with no self edges (i.e. no loop of length $1$) to have an invertible adjacency matrix?
In this case, the adjacency matrix is symmetric (i.e. $A = A^\top$). Moreover, all the…
the_candyman
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Interpretation of Symmetric Normalised Graph Adjacency Matrix?
I'm trying to follow a blog post about Graph Convolutional Neural Networks. To set up some notation, the above blog post denotes a graph $\mathcal{G}$, it's adjacency matrix $A$, and the degree matrix $D$.
A section of that blog post then says:
I…
aaronsnoswell
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Proof - raising adjacency matrix to $n$-th power gives $n$-length walks between two vertices
I came across the formula to find the number of walks of length $n$ between two vertices by raising the adjacency matrix of their graph to the $n$-th power.
I took me quite some time to understand why it actually works. I thought it would be useful…
BMBM
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Spanning forests of bipartite graphs and distinct row/column sums of binary matrices
Let $F_{m,n}$ be the set of spanning forests on the complete bipartite graph $K_{m,n}$. Let $$S_{m,n} = \{(r(M), c(M)), M \in B_{m,n} \}$$ where $B_{m,n}$ is the set of $m \times n$ binary matrices and $r(M), c(M)$ are the vectors of row sums and…
Jair Taylor
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Who first noted that entries in the powers of an adjacency matrix of a graph count the number of walks on the graph?
The Wikipedia article on powers of an adjacency matrix presently (as of 2022) notes the neat combinatorial fact that, given an adjacency matrix $A$ of some graph, entries of the $n$th power of the adjacency matrix, $A^n_{ij}$, count the number of…
Mark S
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How to tell if a directed graph is acyclic from the adjacency matrix?
Suppose you have an adjacency matrix $A$ for a directed graph $G=\{V,E\}$, so $A_{ij} = 1$ if $V_i\rightarrow V_j \in E$, and $A_{ij}=0$ otherwise. Many properties of the graph can be derived from this adjacency matrix. For instance, $(A^n)_{ij}$…
David Pfau
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Why can powers of the adjacency matrix determine connections in the graph?
I am studying graph theory and I am not sure about how the power of the adjacency works. I know that the $k$-th powers of $A$ tell us about connections in the graph, and I can read lengths between a vertex to vertex after taking powers of the…
Uka
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What can we say about two graphs if they have similar adjacency matrices?
Suppose we have two (finite, simple, undirected) graphs, what can we say about these graphs if they have similar adjacency matrices?
Observations to begin with:
If $G_1$ and $G_2$ are isomorphic, then they have similar adjacency matrices, $A_1$ and…
Rebecca J. Stones
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2 answers
Reorder adjacency matrices of regular graphs so they are the same
Given a matrix A of a strongly $k$ regular graph G(srg($n,k,\lambda,\mu$);$\lambda ,\mu >0;k>3$). The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$.
$A_x$ is the symmetric matrix of the graph $(G-x)$, where $C$…
Michael
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7
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Vertices in a graph with the same number of closed walks
A graph is called walk regular if the number of closed walks starting from vertex $u$ of length $k$ does not depend on $u$. If $A$ is the adjacency matrix of the graph, this means that $A^k$ has equal diagonal values for every $k>0$.
I'm interested…
askarmboze
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1 answer
Intuition of the relationship between laplacian matrix, adjacency matrix and degree matrix
I was studying graph theory and came across this tutorial. In page 13 it said
The Laplacian matrix $L$ is equal to $D-A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. I am having a hard time to understand the intuition of how…
user3667089
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1 answer
How many upper triangular $(0,1)$ matrices are equivalent under a permutation of rows?
Let $M_{n}$ be the set of $n \times n$ upper triangular (0,1) matrices with at least one $1$ in every column. $M_{n}$ includes the following matrices
$$ \left\{ \begin{bmatrix}1&1&1\\0&1&1\\0&0&1\end{bmatrix}, \quad…
scarf
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Finding a symmetric adjacency matrix closest to a given (non-symmetric) adjacency matrix
I am trying to solve a problem on graphs, which I have reduced to the following optimization problem in matrix $X \in \{0,1\}^{n \times n}$
$$\begin{array}{ll} \text{minimize} & \| X - A \|_F^2\\ \text{subject to} & X 1_n = m 1_n\\ &…
Jonathan
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2 answers
Obtaining the adjacency matrix of Cayley graphs
Is it possible to obtain the adjacency matrix of a Cayley graph of $Z_3 \times Z_5$? (Manually or by using a software like GAP).
Will there be a pattern for adjacency matrices of Cayley graphs for a particular type of groups considered (i.e. if we…
