Questions tagged [spectral-graph-theory]
Questions related to the spectrum of graphs, defined using one of the possible variants of the discrete Laplace operator or Laplacian matrix. See https://en.wikipedia.org/wiki/Discrete_Laplace_operator
431 questions
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Spectraly extremal connected trees, a conjecture on majorisation of Laplacian eigenvalues
The precise conjecture that I want to prove or disprove is the following:
Let $P_n$ denote the path graph with n vertices, $S_n$ the star graph with n vertices, and $T_n$ an arbitrary connected tree ...
1
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0
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67
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What are some consequences of Graph eigenvectors delocalization?
Given a $d$-regular graph $G$ on $n$ vertices, suppose that every normalized
eigenvector $v$ of the adjacency matrix $A$ satisfies a strong
delocalization bound
$$
\|v\|_\infty \;\ll\; \frac{\log n}{...
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1
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1
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122
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Intersection of a set and a sphere on a Ramanujan graph
Let $G$ be a $d$-regular Ramanujan graph on $n$ vertices.
Fix a subset of vertices $S \subseteq V(G)$, and let $v_0$ be a uniformly random vertex in $G$.
I'm interested in the expected number of ...
- 387
1
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0
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122
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On the largest Eigenvalue of a certain "graph Laplacian"
I am interested in the following problem, which came up while working on this paper about estimating Betti numbers of Kähler manifolds, where we were not able to solve it and had to resort to ...
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1
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1
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Graph-theoretic conditions leading to singularity of adjacency matrices
Let $G$ be a directed graph on $n$ vertices, with adjacency matrix $A \in \{0,1\}^{n \times n}$ (loops allowed). It is well-known that if one row of $A$ is a real linear combination of other rows, ...
- 4,150
19
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2
answers
1k
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Off-diagonal entries of $(AA^T)^{1/2}$ for the bipartite adjacency matrix of a tree
Let $T$ be a tree with a bipartition of its vertices into sets $X = \{v_1, \dots, v_m\}$ and $Y = \{w_1, \dots, w_n\}$. Define the $m \times n$ bipartite adjacency matrix $A$ by
$$ A_{ij}=\begin{cases}...
- 4,503
4
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0
answers
52
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Determinant Spectrum of Laplacians for Connected graphs
Are there formulas or computational results that show the spectrum of sub-determinant values of Laplacians for connected graphs? Specifically, if we have $N$ vertices and we have the set of all ...
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52
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Spectral Clustering Algorithms Requiring Operator Diagonalizability
Let us focus on the category of directed graphs whose adjacency matrix (or random walk operator) is diagonalizable. Is there any spectral clustering algorithm that is specifically designed to operate ...
- 4,150
0
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0
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53
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Peer reviewed works relating spectral graph theory and dynamical systems through piecewise isometries (PWIs)
While Piecewise Isometries (PWIs) represent an area of active research for their insights into complex orbits of dynamical and fractal systems, the study of holonomy-twisted Ihara zeta functions has ...
- 621
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128
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second largest eigenvalue and phase transition of graphs
I study (several) families of finite, directed graphs $(G_n)_{n\ge 0}$, where $G_0$ is "very complicated" and $G_n$ has no edges at all for $n$ large enough; in between, there appears to be ...
- 2,359
3
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2
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499
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Clique number of Cayley graph on finite field
Let $k$ be a finite field and $S\subset k^\times$ a subgroup containing $-1$ (in particular $S$ is cyclic). Consider the Cayley graph $G=\operatorname{Cay}(k,S)$, i.e. the graph whose vertex set is $k$...
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90
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Transfer of spectral data from stratified surfaces to embedded graphs
Let $\Gamma \subset X$ be a finite connected $(q+1)$-regular Ramanujan multigraph, embedded as the 0,1-skeleta of a stratified (smooth away from $\Gamma$) 2-complex $X \subset \mathbb{R}^3$.
A ...
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2
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1
answer
141
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Twisted zeta functions and automorphism-induced periodic dynamics of Ramanujan graphs
Let $\Gamma$ be a finite connected $(q+1)$-regular graph with fundamental group $\pi_1(\Gamma)$, and let $\mathrm{Char}_{\Gamma}:=\mathrm{Hom}(\pi_1(\Gamma),U(1))\cong (S^1)^r$ denote the compact ...
- 621
2
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1
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209
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How to estimate differences between pairs of numbers with constraints
Say we have $N$ unique numbers, and we want to estimate the differences between all $^NC_2$ unique pairs from the $N$ numbers, but we don't care about the absolute values of these numbers. The average ...
- 163
2
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0
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85
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Dynamical systems of twisted Ihara zeta functions on graphs
A twisted Ihara zeta function of a finite connected graph $\Gamma$ is sensitive to a quantity known as holonomy. The zeta function can be defined as:
$$\zeta_{\Gamma}(u, \rho) := \prod_{[P]} \left(1 - ...
- 621
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0
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148
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Product of combinatorial Laplacian eigenvalues is an integer
From Kirchhoff matrix tree theorem we know that the number of spanning trees in a graph is equal to the product of the combinatorial Laplacian eigenvalues (removing eigenvalue 0) divided by the number ...
- 549
18
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2
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1k
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Tree with $1+\sqrt{2}$ an eigenvalue of its adjacency matrix
What is an example of a tree with $1+\sqrt{2}$ an eigenvalue of its adjacency matrix?
Such a tree must exist since "Every totally real algebraic integer is a tree eigenvalue", Justin Salez,...
- 2,139
3
votes
2
answers
243
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A variant of the vertex clique cover problem
Assume that I have a graph $G(V, E)$, where each vertex $v$ is assigned a non-negative weight $w(v) \geq 0$. I aim to find a partition of the graph into cliques $C_1, C_2, \ldots, C_k$ (Here, $k$ is ...
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1
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Is there any way to find Steklov eigenvalues of graph with the help of certain kind of metric in spectral graph theory?
From Fan R.K. Chung's book Spectral graph theory (Regional Conference Series in Mathematics. 92. Providence, RI: American Mathematical Society (AMS), pp. xi+207 (1997), ISBN:0-8218-0315-8, MR1421568, ...
4
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1
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199
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Spectrum of a periodic directed triangular lattice in the large $N$ limit
I am interested in whether there is a closed-form, or even approximate, result for the distribution of eigenvalues of the adjacency matrix of a periodic directed triangular lattice as the size of the ...
- 141
4
votes
1
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711
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Minimum eigenvalue of a symmetric matrix
I was solving a problem and got stuck on the following:
Let $[p] = \{1, \ldots, p\}$ where $p \in \mathbb{N}$. Let $P(n, r)$ denote the set of all injective functions from $[r]$ to $[n]$ and write a ...
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4
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1
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312
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What is the resistance between two vertices on the Hanoi-towers graph?
The Hanoi graph $H^n_k$ is the graph with nodes representing states of a Hanoi puzzle with $n$ discs and $k$ pegs, and edges representing the various moves of the discs from peg to peg.
The Hanoi ...
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1
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0
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138
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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 ...
- 4,150
3
votes
0
answers
86
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Is this bipartite equivalent of 1-walk-regular graphs known?
A graph $G$ is 1-walk-regular if
for each vertex $v$ the number of closed walks of length $\ell$ starting (and ending) at $v$ depends only on $\ell$ but not on $v$.
for each edge $vw$ the number of ...
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0
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0
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55
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Change in two spectral deviations due to edge deletion in a signed graph
Prove (or disprove) the following. Let $\Sigma=(G,\sigma)$ be a given signed graph. If $\lambda_1\ge\lambda_2\ge\cdots\ge \lambda_n$ and $\mu_1\ge\mu_2\ge\cdots \ge \mu_n$ are the eigenvalues of the ...
- 61
1
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0
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253
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Eigenvalues and eigenvectors of the path Laplacian
Consider the Laplacian matrix of the path graph:
$$
L = \begin{bmatrix}
1 & -1 & 0 & \cdots & 0 & 0\\
-1 & 2 & -1 & \cdots & 0 & 0\\
0 & -1 & 2 & \...
- 11
0
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1
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150
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Computing spectrum of very simple Schrödinger operator
I asked this question recently on a thread in math stack exchange, but with no real answers suggested. I think this is a relatively simple variation on the classical free Laplacian spectrum, so I ...
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4
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2
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338
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Multiplicity of the smallest non-zero Laplacian eigenvalue for tree graphs
How accurate is the following statement: "For tree graphs, the multiplicity of the smallest non-zero eigenvalue $\lambda_2$ of the Laplacian is 1." If not valid, in which cases does it fail ...
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4
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1
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168
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When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?
Question: Let $G$ be a strongly connected directed graph on $n$ vertices with Laplacian $L(G)$. Then $L(G)$ has one zero eigenvalue $\lambda_1=0$ and $n-1$ nonzero eigenvalues $\lambda_2,\ldots,\...
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1
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How to prove two non-isomorphic strongly regular graphs are not Seidel switching isomorphic?
It is known that $L(K_{4,4})$ and the Shrikhande graph are two non-isomorphic strongly regular graphs but they are Seidel switching isomorphic.(Cvetknvic-Rowlinson-Simic An Introduction to the Theory ...
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4
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2
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282
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Existence of disjoint expanders in a graph
Let $n$ be some sufficiently large positive integer and let $V$ be a set of $n$ vertices. Are there always disjoint sets of edges $E_1,\ldots, E_{\log n}$ such that $G_i = (V,E_i), \forall i = 1,\...
- 163
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214
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Spectra of Coxeter diagrams and representations of Coxeter groups
Let $S$ be a Coxeter diagram, i.e. an unoriented graph, edges labelled by weights $3,4,5,...$ . This includes (affine) Dynkin diagrams,
Then
Theorem 3.1.3 of Brouwer and Haemers' Spectral Graph ...
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1
answer
167
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Function of eigenvalues of Laplacian matrix
Let $G$ be a simple $n$-vertex graph and let $\mu_n\geq\mu_{n-1}\geq\dots\geq\mu_1$ be the eigenvalues of its Laplacian matrix, how can I find a function $$f(\mu_1,\mu_2,\dots\mu_n) \text{ such that } ...
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3
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0
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Inverse of adjacency matrix of overlapping cycle graphs?
Consider a graph composed of two overlapping cycles: one cycle of length $\ell$ and one cycle of length $m$ where the two cycles share $e$ edges. See picture as an example:
The eigenvalues and ...
- 304
0
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1
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Spectral theory: a key to unlocking efficient insights in network datasets
In the context of directed or undirected graphs, matrices such as adjacency and Laplacian matrices are commonly used. The eigenbasis of these matrices addresses some practical implications, such as ...
- 4,150
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0
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241
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A reference for high girth expander graphs
I'm looking for a reference to the existence of family of constant (or bounded by constant) degree graphs which are both high-girth (meaning the ratio girth/diameter is uniformly bounded from below by ...
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3
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951
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Real-world examples of unweighted directed graphs
Social Networks, Internet Traffic, Citation Networks, Transportation Networks, and Biological Networks exemplify real-world instances of unweighted directed graphs. Within these domains, the Graph ...
- 4,150
1
vote
2
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299
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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 $...
- 4,150
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0
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142
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How can I measure similarity between two graphs with identical topology but different edge weights
I have two graphs, G1 and G2, with exactly the same topology. Their only difference lies in the edge weights, which vary between 0 and 1.
How can I measure the similarity between G1 and G2 under these ...
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2
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1
answer
146
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Testing for equal characteristic polynomials using a single determinant calculation
Let $A_1,A_2$ be $n\times n$ symmetric matrices over $\{0,1\}$, and let $p_1, p_2$ be their respective characteristic polynomials over the rationals.
If $p_1 \ne p_2$, then there is some positive ...
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3
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1
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Some questions about induced subgraphs of the directed hypercube graph
Let $Q^n$ be the hypercube graph in $n$ dimensions. Hao Huang famously showed that any induced subgraph on more than $2^{n-1}$ must have maximum degree $ \geq \sqrt{n}$. It is also known that this ...
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0
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137
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Given a low-rank symmetric positive semidefinite matrix and a basis of its nullspace, is there a fast way to get the nonzero eigenvalues?
I have a (possibly dense) $k\times k$ real matrix $L = AA^T + B^T B$, a type of combinatorial Laplacian (self-adjoint, symmetric, positive semidefinite) of rank $(k-n)$ and possibly repeated nonzero ...
- 21
2
votes
1
answer
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Invertibility of message passing with invertible parametrization
Consider the message passing framework defined by, $$f(\boldsymbol{x}_i)= \boldsymbol{x}_i + \sum_{j \neq i} (\boldsymbol{x}_i -\boldsymbol{x}_j) g(\|\boldsymbol{x}_i -\boldsymbol{x}_j\|^2),$$ for $i=...
2
votes
0
answers
168
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Bound on the magnitude of the entries of the Laplacian pseudo-inverse
Let $L$ be the laplacian matrix of a connected graph $G$ with real positive weights and $N$ vertices, or that can be assumed to have binary weights for simplicity.My goal is to bound $\Vert L^+\Vert_{\...
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3
votes
0
answers
119
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Second eigenvalue of primitive matrix
Let $A$ be a primitive $N\times N$-matrix with positive entries, that is there is $n>0$ such that $(A^n)_{i,j}>0$ for all $i,j$. For brevity, assume the entries consist only of $0$ and $1$.
The ...
- 153
0
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1
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123
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Number of bi-directional (or symmetric edges) [closed]
I am trying to figure out the least number of directed edges that would be bi-directional after constructing a graph with $2k-1$ nodes that are each $k$ in-degree. For example, $2(2)-1=3$ nodes that ...
- 1
2
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0
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83
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Regularize a graph while embedding the spectrum of adjacency matrix
Given an irregular graph $G$ whose maximum degree is $d$, I am interested in producing a new graph $G'$ which is regular and has the spectrum of the adjacency spectrum of $G$ embedded in the spectrum ...
- 21
1
vote
0
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143
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Diameters of random bipartite graphs
Given two partite sets of vertices $U$ and $V$ of size $n$. Each vertex in $U$ uniformly randomly selects $K$ ($K$ is a constant and $K\ll n$) vertices in $V$ without replacement and connects a ...
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2
votes
1
answer
386
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Do balls in expander graphs have small expansion?
Consider a $d$-regular infinite transitive expander graph $G$, and let $B_r$ be a ball of radius $r$ in $G$. Can one place any upper bounds on the expansion of $B_r$?
My intuition is that $B_r$ will ...
- 31
2
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0
answers
375
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Max-cut from Laplacian
(This question seems like very standard material for those well-versed in the subject. I thought I would get a quick answer from Math stackexchange, but to no avail.)
Given a weighted graph with $n$ ...
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