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

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A new lower bound for the chromatic number of a graph?

Let $S_{+}(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of a graph $G$. Let $S_{-}(G)$ denote the sum of the squares of the negative eigenvalues and $q$ the ...
Clive elphick's user avatar
11 votes
0 answers
549 views

Ramanujan Digraphs?

In Gowers' paper on quasirandom groups, he suggests a spectral theory of bipartite graphs employ the singular values of the bipartite adjacency matrix. Accordingly, singular values appear to be a ...
mdevos's user avatar
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10 votes
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216 views

Cospectral mate of rhombic dodecahedron

I am wondering if the following pair of cospectral graphs was previously known. The rhombic dodecahedron graph looks like this (graph6 string: 'M?????rrAiTOd_YO?'): As far as I know, it was previously ...
David Roberson's user avatar
10 votes
1 answer
548 views

Are there two-sided $\varepsilon$-expanders with independent sets of size $(1-\varepsilon)n$?

Terry Tao's notes on expander graphs has the following exercise: Exercise 13 Let $G$ be a $k$-regular graph on $n$ vertices that is a two-sided $\epsilon$-expander for some $n > k \geq 1$ and $\...
JeremyKun's user avatar
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8 votes
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Explicit constructions of Ramanujan graphs

I am trying to find a list of all the explicit constructions of Ramanujan graphs. By a Ramanujan graph I mean a $k$-regular multi-graph $G$ such that all the non-trivial eigenvalues $\lambda$ of the ...
richarddedekind's user avatar
8 votes
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122 views

Conceptual explanation for the gap in the spectrum of biregular trees

Ramanujan graphs are defined as $(q+1)$-regular graphs for which all nontrivial eigenvalues of the adjacency operator are contained in the interval $$[-2\sqrt{q}, 2\sqrt{q}].$$ The reason for this ...
Antoine Labelle's user avatar
8 votes
0 answers
391 views

Bounding eigenvalues by taking high powers of matrices: history?

Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that $$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$ for any $k\geq 1$. ...
H A Helfgott's user avatar
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Is there a version of Weyl's law for graph Laplacians?

Is there a version of Weyl's law or a local Weyl's law for eigenvectors of the graph Laplacian? For some context, a colleague in statistics has encountered eigenvectors of the Laplacian for certain ...
Gabe K's user avatar
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7 votes
0 answers
193 views

Surprising symmetry in the Ramanujan bound

The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy $$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$ With a ...
Antoine Labelle's user avatar
7 votes
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302 views

When can the Cheeger constant be well-approximated by ``Hamming balls''?

Given a graph G, the Cheeger constant is defined by $$ \DeclareMathOperator{\Vol}{Vol} h_G := \min_{S \subseteq V, \Vol S \leq (\Vol G)/2} \frac{|\partial S|}{\Vol S}. $$ Here, $\Vol S$ is the sum of ...
Eric Tressler's user avatar
5 votes
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115 views

The Smith decomposition of the graph Laplacian and Locality

Let $X$ be a graph. Let $V(X)$ and $E(X)$ be the sets of vertices and edges of the graph respectively. If $f:V(X) \rightarrow G$ where $G$ is an abelian group, then one can define a graph Laplacian as ...
nabil's user avatar
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(How) does the spectral gap of the $n\times n$ Rubik's cube close with $n$?

Consider the spectrum of the adjacency matrix $A$ of the Cayley graph of the standard, 3x3x3 Rubik's cube generated with the usual quarter-turn and half-turn twists of each face (the Singmaster ...
Mark S's user avatar
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Closed paths, closed trails and traces

Let $A$ be the adjacency matrix of a (non-oriented) graph $\Gamma$. Then $\textrm{Tr} A^k$ equals both the sum $\sum_i \lambda_i^k$ of $k$th powers of eigenvalues of $A$, on the one hand, and the ...
H A Helfgott's user avatar
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Can the corollary of the Ihara–Bass formula be extended to $ u^2 = 1 $?

Suppose there is a finite undirected graph $G(V,E)$ having $n$ vertices and $m$ edges. The non-backtracking matrix $B$ is indexed by $2m$ directed edges and defined as $$ B(a \to b, c \to d) = \delta_{...
Eli4ph's user avatar
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spectrum of orthogonality graphs

The orthogonality graph $\Omega(n)$ with $2^n$ vertices is the graph with vertex set $\{-1,+1\}^n$, with two vertices being adjacent if and only if they are orthogonal (as vectors in the standard ...
Clive elphick's user avatar
5 votes
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258 views

(Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples

Let $G = PSL(2,q)$. I'm interested in the Cayley graphs of $G$ generated by triples $(A,BAB^{-1},B^{-1}AB)$, where $A, B \in G$ are elements of order $2, 3$ respectively: such a triple generates all ...
Robin Saunders's user avatar
4 votes
0 answers
132 views

Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root

Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...
baronbrixius's user avatar
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157 views

Can resolution of the Kadison-Singer Problem provide progress on the Komlos Conjecture?

This is not a concrete question, just some thoughts. The Komlos Conjecture is as follows- There exists an absolute constant $C>0$, such that the following holds: For all $d$ and any set of vectors ...
Sudipta Roy's user avatar
4 votes
0 answers
151 views

Relation between two conjectures on reconstruction of graphs

In spectral graph theory, there is a conjecture that claims: Almost every graph is determined by its adjacency spectrum ($DS$). This conjecture belongs to professor Willem Haemers. Also, we have a ...
Shahrooz's user avatar
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474 views

Laplace spectra of "half" grid graph

Let $G=(E,V)$ be a simple graph. The graph Laplacian is given by $$ L= D-A,$$ where $D$ is the degree matrix (diagonal matrix with entries corresponding to the degree of the vertex) and $A$ the ...
ortofoxy's user avatar
4 votes
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139 views

algebraic connectivity of a tree

Suppose that $T$ is a tree with $n$ vertices and $L$ is the Laplacian matrix of $T$ and $0=\mu_1 \leq \mu_2 \leq \cdots \leq \mu_n$ are laplacian eigenvalues. I think the multiplicity of $\mu_2$ can ...
MH.Fakharan's user avatar
4 votes
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651 views

Counting loops in degree: 1 or 2?

Here's what seems to be an annoying technicality when dealing with loops in graphs. In the literature on expander graphs (and surely not only), it seems to be the convention that a loop at vertex $v$ ...
amakelov's user avatar
  • 987
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124 views

An inequality from the "Interlacing-1" paper

This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132) For the argument to ...
InterlacingStudent's user avatar
4 votes
0 answers
589 views

The Bilu-Linial conjecture and Ramanujan graphs

The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...
user6818's user avatar
  • 1,883
4 votes
0 answers
181 views

Relaxation = absorption?

Let $A$ be a stochastic matrix, that is, the entries are non-negative and each row adds to $1$. Assume that it is primitive, that is, $A^n$ has only positive entries for sufficiently large $n$. We ...
anon's user avatar
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61 views

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 ...
Curious's user avatar
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3 votes
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64 views

Clique number and spectrum of a graph

In the Wikipedia article on Grassmann graph it is stated that in this graph: $$\omega=1-\frac{\lambda_{max}}{\lambda_{min}}$$ where $\omega$ is the clique number of the graph, and $\lambda_{max}$ and $...
Ghodrati's user avatar
  • 175
3 votes
0 answers
182 views

Spectral norm and "operator norm" for hypergraphs

Consider a $d$-regular, $k$-uniform hypergraph: the elements $S$ of its set $E$ of edges are subsets of $V$ of size $k$, and each vertex $v\in V$ is in $d$ edges. We can then define its adjacency ...
H A Helfgott's user avatar
  • 19.3k
3 votes
0 answers
250 views

Eigenvalue bounds and triple (and quadruple, etc.) products

Very basic and somewhat open-ended question: Let $A$ be a symmetric operator on functions $f:X\to \mathbb{R}$, where $X$ is a finite set. Assume that the $L^2\to L^2$ norm of $A$ is $\leq \epsilon$, i....
H A Helfgott's user avatar
  • 19.3k
3 votes
0 answers
141 views

Chromatic number of regular graphs using spectra

There exist inequalities relating the maximum and minimum eigenvalues of the adjacency matrix of a graph with its chromatic numbers, i.e. the Wilf's and Hoffmann's inequalities, which put together ...
vidyarthi's user avatar
  • 2,007
3 votes
0 answers
141 views

How related are Fourier transforms on finite groups and Fourier transforms on graphs?

Here are two generalizations of the notion of a Fourier transform. I am also aware of the Pontryagin Duality generalization for locally compact abelian groups, though I am personally more concerned ...
Jackson Abascal's user avatar
3 votes
0 answers
823 views

Principal eigenvector of non-negative symmetric block matrix is approximated by a linear combination of the principal eigenvectors of the blocks

Let $ M \in \mathbb{R}^{n \times n} = \begin{bmatrix} A & B \\ B^T & C \end{bmatrix} $ for some nonnegative $A \in \mathbb{R}^{k \times k}, B \in \mathbb{R}^{k \times n-k}, C \in \mathbb{R}^{n-...
Ross Griebenow's user avatar
3 votes
0 answers
234 views

Does the zeta regularized Laplacian determinant measure the volume of some parameter space? How many "spanning trees" on a manifold?

Let $(M,g)$ be a Riemannian manifold, with Laplacian $\Delta$. If $\lambda_i$ are the nonzero eigenvalues of $\Delta$, we can define the zeta function $\zeta(s) = \Sigma \lambda_i^{-s}$. By analytic ...
Elle Najt's user avatar
  • 1,422
3 votes
0 answers
116 views

The algebraic connectivity of graphs with large isoperimetric number

Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$. The isoperimetric number of $G$, denoted $i(G)$, is defined by $$i(G) = \min_{|S| \leq |V|/2} \frac{e(S,\bar{S})}{|S|},$$ where $e(...
j.s.'s user avatar
  • 519
3 votes
0 answers
91 views

Analogues of relative property $(\tau)$ for Schreier graphs

Suppose I have an expanding family of Schreier graphs $Z_n=\text{Sch}(G_n,S_n,X_n)$ of groups $G_n=\underbrace{G\wr\ldots\wr G}_{\text{$n$ times}}$ acting on sets $S_n=S^n$ by generating sets $X_n$, ...
amakelov's user avatar
  • 987
3 votes
0 answers
41 views

About the partial expectation polynomials in the Interlacing-I paper and perfect matchings

I am thinking of the polynomials $f_{s_1,s_2,..,s_k}$ as in the definition 4.3 in this paper http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf In the use of these ...
InterlacingStudent's user avatar
3 votes
0 answers
253 views

About the small set expansion conjecture

Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - \...
user6818's user avatar
  • 1,883
3 votes
0 answers
233 views

Must distinct tree eigenvalues be relatively far apart?

How close to each other can two distinct eigenvalues of a tree be, as a function of the number $n$ of nodes ? For example, the path $P_n$ exhibits a gap of order $\frac{2\pi^2}{n^2}$ asymptotically ...
Dr J's user avatar
  • 145
3 votes
0 answers
799 views

A problem on graph theory and complex numbers!

Let ${\mathcal G} = ({\mathcal V},{\mathcal E})$ be a simple connected undirected graph with $n$ vertices. Also let $z_1, \ldots, z_n \in {\mathbb C}$ be complex numbers such that $$ ||z_1||=\ldots = |...
Mohammad Khosravi's user avatar
3 votes
0 answers
127 views

Effect of removing a Hamiltonian cycle on the Laplacian spectrum

Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$). Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$. ...
Felix Goldberg's user avatar
3 votes
0 answers
155 views

inverse M-matrix times mixed-sign vector

Recently a colleague and I came across this unusual phenomenon. Take $M\in\mathbb{R}^{n\times n}$ a singular irreducible M-matrix, and $b\in\mathbb{R}^{n}$ such that the system $Mx=b$ is solvable (so,...
Federico Poloni's user avatar
2 votes
0 answers
38 views

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 ...
BenJones's user avatar
2 votes
0 answers
72 views

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_{\...
sd24's user avatar
  • 21
2 votes
0 answers
39 views

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 ...
Chaithanya's user avatar
2 votes
0 answers
118 views

Graph Laplacians, Riemannian manifolds, and object collisions

To preface this question, I am a part-time game developer and full-time optimization fiend. I am working on object collisions at the moment and many resources I have found online are more-or-less just ...
HeyoItsMateo's user avatar
2 votes
0 answers
303 views

Spectral norm bound for lower triangular matrix

Let $A$ be a $0/1$ square matrix which can be permuted to a non singular or a singular lower triangular matrix. Determinant is either $0$ or $1$. Can we provide tighter upper bounds on its spectral ...
Turbo's user avatar
  • 13.6k
2 votes
0 answers
113 views

Formulas to determine the value of graph energy with addition or deletion of edges

If $G$ is a graph, then the graph energy of $G$ denoted by $E(G)$ is defined as the sum of absolute values of eigenvalues of the adjacency matrix of $G$. It is known that $E(G)\geq E(G-v)$, where $ ...
gete's user avatar
  • 203
2 votes
0 answers
158 views

Closed geodesics and eigenvalues in a non-regular graph

Let $\Gamma$ be an undirected graph the degree of whose $n$ vertices is $\leq D$ without necessarily being constant. Say we have bounds of type $\leq \gamma^{2 k}$ for the number of closed geodesics ...
H A Helfgott's user avatar
  • 19.3k
2 votes
0 answers
180 views

The rank of a Laplacian-type matrix

Suppose that $M$ is an integer, symmetric matrix of order $n>2$ with the positive integers $K_1,\dotsc,K_n$ on its main diagonal, and with all the off-diagonal elements equal to $0$ or $1$ so that ...
Seva's user avatar
  • 22.8k
2 votes
0 answers
191 views

Expansion of random subgraphs of a bi-regular bipartite graph

Let $G = (L, R, E)$ be a bi-regular bipartite graph, with $|L|=n$ and $|R| = C \cdot n$, where $C$ is a large constant. Let $d$ be its (constant) right-degree. We know $G$ is a good spectral expander. ...
Daniel86's user avatar
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