All Questions
Tagged with directed-graphs spectral-graph-theory
9 questions
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PageRank in directed graphs: equivalence of iterative and eigenvalue methods
Given a directed graph $ G $ with $ n $ nodes, we can represent this graph using an adjacency matrix $ A $. The stochastic matrix $ S $ can be derived from the adjacency matrix using the following ...
- 4,058
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36
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A class of directed graph, when their minimal polynomial of the adjacency matrix matches the characteristic polynomial
We consider an unweighted directed simple graph, $G$, with a Hamiltonian cycle.
Q. Assume that the adjacency matrix of $G$ is non-singular. Do the characteristic and minimal polynomials of the ...
- 4,058
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2
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198
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Topology of directed graph $G$ with non-singular adjacency matrix
Given a directed graph $G$ with non-singular adjacency matrix,
Q. Is there a directed
subgraph $H$ in $G$ that can be represented as the union of disjoint cycles such that $H$ contains all nodes of $...
- 4,058
1
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143
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Eigenvalues of directed graph with one outward edge for each vertex
I am concerned with unweighted directed graphs where each node contains exactly one edge pointing to another node, which could be itself. In other words, each row of the adjacency matrix contains one ...
- 283
4
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207
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Digraphs with unique walk of length $k$ between any two vertices
Let $G$ be a digraph such that there is an unique directed walk of length $k$ between any two vertices.
Equivalently, if $A$ is the adjacency matrix of $G$, then $A^k$ is the matrix with all entries $...
- 3,446
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Digraphs with same number of semiwalks
This is a follow-up question to Characterisation of walk-equivalent digraphs.
Question: Do there exists two directed graphs $G$ and $H$ consisting of the same number ($n$) of vertices, such that
\...
- 493
1
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138
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Characterisation of walk-equivalent digraphs
Setting Let $G=(V,E)$ be an undirected graph. A walk $\pi$ in $G$ of length $k$ is a sequence of $k+1$ vertices $v_1,\ldots,v_{k+1}$ such that for each $i\in[1,k]$,
$\{v_i,v_{i+1}\}\in E$. Let $H=(W,F)...
- 493
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1
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Two cospectral (normal) digraphs which are not orthogonal similar
Preliminaries
A complex matrix $A$ is normal when $A$ and $A^*$ commute. A real matrix $A$ is normal when $A$ and $A^t$ commute.
Two complex matrices $A$ and $B$ are said to be unitary similar if ...
- 493
7
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4
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Minimum negative eigenvalue of zero-one matrices
The following question must have been answered decades ago.
For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...
- 4,679