**3**

votes

**1**answer

212 views

### Meaning of eigenvalue 1 and symmetry in Laplacian spectra of graphs

We often see normalized Laplacian spectra of graphs where density on eigenvalue 1 serves as an axis of symmetry, with particularly high (blue spectra in the figure) or low densities (red spectrum) ...

**0**

votes

**2**answers

120 views

### Form of the Shannon capacity for Heptagon?

Is the $0$-error capacity of $7$-cycle:
$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?

**4**

votes

**2**answers

169 views

### eigenvalue estimate of the adjacency matrix

The adjacency matrix of a nonempty (undirected) graph has a strictly positive largest eigenvalue $\lambda_\max$. A very easy upper estimate for it can be obtained directly by Gershgorin's theorem:
$$
...

**5**

votes

**0**answers

179 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 ...

**8**

votes

**1**answer

140 views

### Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...

**4**

votes

**2**answers

186 views

### Largest eigenvalue adjacency matrix-link deletion

Let G be a connected undirected graph and G\e be a graph obtained by removing a random link e from the graph G. Let $\lambda_1(A(G))$ be the largest eigenvalue of the adjacency matrix of graph G. Is ...

**8**

votes

**1**answer

233 views

### A conjecture about strongly regular graphs

Let $G \neq K_{v}$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$
So far I have ...

**5**

votes

**1**answer

197 views

### Full-rank factorization of the graph Laplacian

Is there a combinatorially meaningful full-rank factorization of the Laplacian matrix of a graph?
The usual factorization $L=BB^{T}$, where $B$ is an oriented incidence matrix, is full-rank if and ...

**3**

votes

**1**answer

168 views

### Cheeger inequality for the maximal eigenvalue

Let $G = (V,E)$ be an undirected graph and let $L = I - D^{-1/2} A D^{-1/2}$ be its normalized Laplacian matrix. The Cheeger inequality asserts that:
$$\frac{\Phi_G^2}{2} \leq \lambda_2 \leq 2 ...

**3**

votes

**1**answer

118 views

### limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?

Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where ...

**5**

votes

**2**answers

421 views

### Roots of matching polynomial of graph

At the end of this preprint, I make the following conjecture concerning the roots of the matching polynomial:
If a graph $G$ is connected and contains a cycle, then the spectral radius of $G$ ...

**8**

votes

**1**answer

213 views

### Spectral lower bounds on the diameter of a graph

There is such a bound, due to Mohar and McKay, using the second-smallest eigenvalue of the Laplacian $\lambda_{2}$:
$$Diam \geq \lceil \frac{4}{n\lambda_{2}} \rceil.$$
This bound is very elegant but ...

**2**

votes

**1**answer

97 views

### Eigenvalues of a graph and its one-edge-delation graph

Let $G$ be any graph with at least one edge and let $e$ be any edge of $G$. Let $G-e$ denote the subgraph of $G$ obtained by deletion of the edge $e$. Assume that $G$ has $n$ vertices.
Suppose ...

**3**

votes

**0**answers

152 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 ...

**1**

vote

**0**answers

78 views

### interpretation of generalized eigenvalue/vectors in spectral graph theory

Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...

**12**

votes

**3**answers

309 views

### How can I prove that a particular family of graphs is integral?

I'm working with an infinite family of graphs that seems to always have all integral eigenvalues, and I'd like to find some way to prove that (if it's true). Call the graphs $G_{n,k}$ and define them ...

**3**

votes

**1**answer

117 views

### The spectral radius of a modified graph

Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...

**0**

votes

**1**answer

133 views

### Which graph topology has the greatest eigenvalue?

I am looking at comparing multiple graph topologies based on their spectra. From the set of all $N\times N$ adjacency matrices, is there any result which points to the adjacency matrix with the ...

**5**

votes

**2**answers

183 views

### Reflexive (hyperbolic) graphs

Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?
By "effective" I mean something ...

**3**

votes

**2**answers

747 views

### spectrum of an adjacency matrix

The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real.
If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few ...

**2**

votes

**1**answer

253 views

### Graph of Grassmannian

Let p be an integer, and let G be the graph $(V=Gr(k,\mathbb{F}_q ^n),E)$
where: $Gr(k,\mathbb{F}_q ^n)$ is the set of all subspace of $\mathbb{F}_q$ of dimension k, and $E=\{ W_1,W_2 \in V | ...

**9**

votes

**1**answer

330 views

### Coherence between different ranking methods of a graph's vertices

Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.
Two natural ways of doing it are:
By the degrees.
By the entries in a ...

**1**

vote

**0**answers

154 views

### If two graphs have same Laplacian spectrums, are they the similar graphs?

If graph A and graph B have exactly the same Laplacian spectrums, can I just say they are same graphs? (only with scaling edge weights and node index reordering) I don't know if there is any explicit ...

**2**

votes

**0**answers

255 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 ...

**1**

vote

**1**answer

303 views

### When does graph Laplacian have eigenvalue -1?

Consider an undirected graph $G$ with (symmetric) adjacency matrix $A \in \{0,1\}^{n \times n}$ and degree sequence $d = (d_i)$ where $d_i = \sum_{j} A_{ij}$. Assume that every node has degree at ...

**7**

votes

**1**answer

360 views

### normalized laplacian spectrum of trees

Is it known for which class of graphs the normalized laplacian has only simple eigenvalues (i.e., with multiplicity one)? In particular, are there trees (or perhaps a specific class of trees) whose ...

**6**

votes

**1**answer

341 views

### Cospectrality and dimension of graphs

Firstly, I apologize if the question is long. I appreciate any helpful answers and ideas.
In the following all graphs are simple and connected.
Let $G$ be graph with vertex set ...

**3**

votes

**2**answers

313 views

### What is the state of the art on triangle-free strongly regular graphs?

From what I've read I've gathered the following facts:
There are seven known such graphs.
Certain parameter sets are ruled out by the Krein conditions and the absolute bound.
Beyond that, little or ...

**6**

votes

**1**answer

357 views

### How to find or constrain “particularly good” (two-sided) spectral expanders?

I'm new to graph theory, but a response to a question I asked a while ago introduced me to the concept of expander graphs.
A k-regular graph (henceforth "graph") on n nodes has eigenvalues k = λ1 ≥ ...

**1**

vote

**2**answers

144 views

### Are trees spectrally determined?

Are trees (connected acyclic graphs) determinable up to isomorphism by their spectra or characteristic polynomials? If not, what other pieces of information may help determine the tree?

**3**

votes

**0**answers

262 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 = ...

**10**

votes

**0**answers

286 views

### Spectral theory of graph Laplacian besides $\lambda_2$

Most of what I've seen about the spectral theory of the graph Laplacian concentrates on $\lambda_2$, the second-smallest eigenvalue. This eigenvalue contains information regarding the connectivity of ...

**3**

votes

**1**answer

246 views

### Graph Laplacian simple eigenvalues

Is there a class of graphs (besides the path graphs) for which we know that the Laplacian L = D - A (where D is the degree matrix and A is the adjacency matrix) has simple spectrum, i.e. all Laplacian ...

**7**

votes

**2**answers

349 views

### Small eigenvalues and spectral clustering

Let $L$ be the discrete Laplacian associated to an undirected graph. It is well-known that the spectral gap of $L$, i.e. the smallest nonzero eigenvalue, is a measure of how well connected the graph ...

**2**

votes

**0**answers

81 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 $C$.
...

**1**

vote

**0**answers

138 views

### Recovering a partition from spectral properties of the graph Laplacian

Let $G$ be a weighted graph with vertices $V$. Let $W$ be its real-valued, non-negative, $|V|\times|V|$ adjacency/affinity matrix. Let $L = \mathrm{diag}(W\mathbf1)-W$ be the (unnormalized) graph ...

**4**

votes

**1**answer

241 views

### What is the largest possible operator norm of a sparse (0,1)-matrix?

Inspired by this question, I was wondering about the following problem:
Consider all $n\times n$ $(0,1)$-matrices with $k$ ones. Which of these matrices has the largest operator norm? And how does ...

**-1**

votes

**1**answer

270 views

### Upper bound on iterations count for power iteration algorithm

I'm stuck trying to get upper bound on iterations count for power iteration algroithm for finding first eigenvalue of adjacency matrix $A$ given tolerance value. I've tried to figure something out ...

**4**

votes

**0**answers

138 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 ...

**0**

votes

**1**answer

298 views

### Random walk on the hypercube

Consider the hypercube $Q_4$. I would like to know how to compute the number of steps of a random walk in this graph such that the probability to be at a vertex is a given number $x$. I think I just ...

**11**

votes

**1**answer

274 views

### Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube.

Consider an integer cube of size $\sqrt{k} \times \sqrt{k} \times \sqrt{k}$, where $k$ is an asymptotically large perfect square number. Place k points in this cube at uniformly random locations, ...

**1**

vote

**1**answer

128 views

### Spectrum of composition of graphs( lexicographic product)

I was wondering how to relate the spectra of the composition of two graphs in term of the factors...someone can help me?

**4**

votes

**1**answer

623 views

### How many distinct eigenvalues does a random graph have?

It is well-known that a random graph a.e. has diameter 2. It is also well-known that the number of distinct eigenvalues of a graph is at least the diameter plus one.
But what is known about the ...

**0**

votes

**0**answers

104 views

### Global solution for spectral clustering

I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...

**10**

votes

**0**answers

401 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 ...

**4**

votes

**2**answers

602 views

### signing a strongly regular graph

Let $A$ be the adjacency matrix of a strongly regular graph. When is it possible to sign $A$ (i.e. replace some of the +1 entries by -1) so that the resulting matrix has exactly two eigenvalues?
I ...

**5**

votes

**1**answer

300 views

### Dimension of Specht Modules $S^\lambda$

In the study of representation theory of $S_n$, we know that the irreducible characters of $\chi_\lambda$ of $S_n$ are indexed by partitions $\lambda \vdash n$. There are several methods in ...

**2**

votes

**1**answer

809 views

### Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries

I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...

**1**

vote

**2**answers

350 views

### How to identify bridge nodes between nearly connected graph components in partitioned adjacency matrices?

I have adjacency matrices which have nearly connected components. That is partitions with a dense number of edges between nodes in the same group and few edges acting as bridges between these groups. ...

**0**

votes

**1**answer

529 views

### Find edge weights that fit given node weights

Let $G = (V,E)$ be a connected simple graph (unweighted, undirected, no selfloops) on $n$ nodes.
Let $\mathbf{d} := (d_1, d_2, ..., d_n) \in \mathbb{R}_{>0}^n$ be a vector of arbitrary given node ...