All Questions
Tagged with spectral-graph-theory co.combinatorics
36 questions with no upvoted or accepted answers
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 ...
- 159
10
votes
0
answers
225
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 ...
- 2,339
8
votes
0
answers
145
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 ...
- 3,446
7
votes
0
answers
304
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 ...
- 1,318
5
votes
0
answers
397
views
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 ...
- 421
5
votes
0
answers
267
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 ...
- 3,612
4
votes
1
answer
303
views
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 ...
- 41
4
votes
0
answers
176
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 ...
- 150
4
votes
0
answers
162
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 ...
- 4,784
4
votes
0
answers
240
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 ...
- 1,462
3
votes
0
answers
61
views
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 ...
- 13.6k
3
votes
0
answers
154
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 ...
- 2,089
2
votes
0
answers
351
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 ...
- 13.9k
2
votes
0
answers
212
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. ...
- 225
2
votes
0
answers
63
views
Antipodal vertices in spectral graph embeddings
Suppose your are given an antipodal graph $G=(V,E)$, that is, for every vertex $v\in V$ there is a unique maximally distant vertex $v'\in V$.
Under which condistions does the following hold:
If $\...
- 13.6k
2
votes
0
answers
112
views
Number of components of self-index complementary graphs
Let $G$ be a simple graph. We say this graph is self-index complementary ($SIC$) if $\lambda_1 (G)=\lambda_1 (\overline{G})$, where $\lambda_1(G)$ denotes the index of the adjacency matrix of the ...
- 4,784
2
votes
0
answers
536
views
On the existence of Graph Monomorphism
A graph monomorphism is an injective graph homomorphism. Determining existence of Graph monomorphism between graph pairs is computationally hard.
Assume we talk only about classes of undirected ...
- 13.9k
1
vote
1
answer
177
views
Spectral characterization of complete or complete bipartite graphs
The Lemma 6 in this paper mention the following spectral characterization of complete or complete bipartite graphs:
Let $G$ be a connected graph with $\ge 2$ vertices. Then $\lambda_2=...=\lambda_{n-...
- 243
1
vote
0
answers
50
views
Bounds on the spectral radius of a perturbed directed graph
Suppose $(G_n)$ is a sequence of strongly connected directed graphs (without multiple edges) with $G_n$ having $n$ edges such that the adjacency matrix $A_n$ of $G_n$ is primitive, and let $(G_n’)$ be ...
- 11
1
vote
0
answers
127
views
Delocalization of eigenvectors of graph Laplacians
Let $(V,E)$ be an undirected, connected graph with $n$ nodes. The graph Laplacian is defined as $L = D - A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. Let $0 = \lambda_1 < \...
- 41
1
vote
0
answers
121
views
Cheeger constant of truncated hypercube
Look at the $d$-dimensional hypercube and truncate it. This means one replaces each vertex by a cycle (of length $d$) in such a way the the new graph is 3-regular.
Question 1: What is the asymptotic ...
- 4,422
1
vote
0
answers
184
views
Large bounded degree expanders in the hypercube
Does the $n$ dimensional hypercube graph contain large bounded degree expanders as subgraphs? For example, of exponential size in $n$?
If not, one could relax the problem and allow the maximum ...
- 2,772
1
vote
0
answers
387
views
Relation between the sum of principal minors of different orders
Let $A$ be a symmetric (0,1)-square matrix of order $n$ having the diagonal entries zero. Let $m$ be the nullity of $A$ (number of zero eigenvalues), denoted by $\eta(A)$. Let $A_1$ be the square ...
- 640
1
vote
0
answers
51
views
Relation between nullity of components to its parent graph
Let $G$ be an undirected graph and the corresponding adjacency matrix be $A$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ are the connected components of the induced graph $G-v$ ( the ...
- 640
1
vote
0
answers
62
views
Cut norm and biclique gap?
Given real $\pm1$ matrix $M\in\Bbb R^{n\times m}$ we have that cut-norm is given by $$\|M\|_C=\max_{\mathcal I\subseteq[n],\mathcal J\subseteq[m]}\Big|\sum_{(i,j)\in\mathcal I\times\mathcal J}M_{ij}\...
- 13.9k
1
vote
0
answers
232
views
Lp norm estimates for the inverse of the Laplacian on a graph
I am looking at a finite connected graph and I would like to know what is the best [i.e. largest] constant $\lambda_p$ in
$$
\sum_x f(x) =0 \implies \| \Delta^{-1} f\|_{\ell^p} \leq \lambda_p^{-1} \|...
- 4,422
1
vote
0
answers
315
views
When does the normalized graph Laplacian have eigenvalue 1?
Let $G= (V,E)$ be a finite, undirected and unweighted graph with $V = \{v_1,\ldots, v_n\}$. Denote by $d_i$ the degree of $v_i$, i.e. the number of vertices that are adjacent to $v_i$. Let $A$ be the ...
- 119
1
vote
0
answers
157
views
Lower bound for the difference between the maximum eigenvalue of a graph with the one of the one-edge-deleted subgraph
I have proposed very recently a question in the following link concerning the title of the current question:
Difference of the maximum eigenvalue of a graph with the one of one-edge-deleted subgraph
...
- 3,738
1
vote
0
answers
104
views
expansion with respect to p-norms for p other than 2
Suppose I have an $d$-regular expander graph with $n$ vertices, where the stochastic version of its adjacency matrix $A$ (with entries $1/d$ and zero) has second eigenvalue $\lambda$.
Let $x \in {\...
- 423
0
votes
0
answers
55
views
Comparing spectral radius of two graphs using the entry of Perron vector
Suppose we have a graph $G$.
Let $A$ be the adjacency matrix of $G$ and $x$ be the corresponding Perron vector.
Let $x = (x_1,x_2,\cdots,x_n)^t$, where $x_i$ corresponds to the vertex $i \in V(G)$.
We ...
- 199
0
votes
0
answers
120
views
Does an extension of the B.E.S.T. theorem for multiple Eulerian circuits exist?
Given a directed multigraph $G=(V,E)$ (multiple edges and loops are permitted) the number of distinct Eulerian circuits for $G$ can be calculated with the B.E.S.T. theorem. Does a similar theory for ...
- 1,295
0
votes
0
answers
53
views
Variation in eigenvalues of adjacency matrices of regular graphs
What is known about the range of spectra of regular graphs? That is, I wish to know the largest intervals in which the minimum and maximum eigenvalues of a graph lie. For example, it is known that the ...
- 2,089
0
votes
0
answers
298
views
How is the second smallest eigenvalue of normalized laplacian bounded for random graphs?
It is well known that for any graph G following holds
$\frac{\lambda_2}{2} ≤ \phi(G) ≤ \sqrt{2\lambda_2}$, where $\phi(G)$ is the conductance of the graph and $\lambda_2$ is the second smallest ...
- 1
0
votes
0
answers
125
views
Lower bounds on the length of circuits, depending on the number of times it crosses itself
I have this problem that I have been stuck on for months, and would like to know if somebody can tell me a way to attack the problem. Let me ask the problem in a simple example below.
Let $G(V,E)$ be ...
- 431
0
votes
0
answers
101
views
4-cycles vs eigenvalue information on quasi-random graphs
My (philosophical) question arises from reading the wonderful paper of Chung-Graham-Wilson where the authors introduces the notion of quasi-random graphs.
The main purpose of the paper is to show ...
- 1,561
0
votes
0
answers
67
views
Singular values and the chromatic number
What relation, if any, is there between the singular values of the adjacency matrix ( or possibly incidence matrix) of a simple graph and its chromatic number. Typically, do we have Hoffmann type, or ...
- 2,089