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4
votes
1answer
209 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 ≥ ...
7
votes
1answer
159 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 ...
3
votes
0answers
76 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 ...
2
votes
1answer
144 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 ...
1
vote
0answers
48 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 ...
2
votes
1answer
73 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 ...
6
votes
2answers
250 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 ...
4
votes
2answers
188 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 $G$ ...
0
votes
0answers
21 views

Diffusion maps for non-Markov

Diffusion maps based on the work of Coifman and Lafon use concepts from Markov chains and heat diffusions. Have there been work to extend diffusion maps to non-Markovian or fractional heat ...
4
votes
1answer
258 views

Graphs which are “distance-regular” with respect to a vertex (but not distance-regular)

A distance-regular graph (DRG) is, in essence, a graph $\Gamma$ of diameter $d$ for which there are integers $c_i, a_i, b_i, (0 \le i \le d)$ such that for all vertices $x$ of $\Gamma$ and for all ...
8
votes
1answer
161 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 ...
0
votes
0answers
90 views

How could I prove this equality for eigenvalues of Laplacian matrix?

I would be glad if you have some comments that how I could prove following statement. Suppose that graph $G =(N, E)$ be given. The the following program computes the $k$-smallest eigenvalues of the ...
2
votes
1answer
74 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
0answers
126 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
0answers
61 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 ...
7
votes
0answers
190 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 ...
12
votes
3answers
279 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
1answer
109 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 ...
4
votes
3answers
591 views

Spectral radius of a proper subgraph

I came across a Chinese reference in the paper "On the spectral radius of trees with fixed diameter" by Guo and Shao. The attribute the following to Q. Li, K.Q. Feng in: "On the largest eigenvalue of ...
0
votes
1answer
119 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
2answers
244 views

Number of trees with the same matching number

Let $\sigma(n,m)$ be the number of trees with $n$ vertices $\{ v_1, \dots, v_n \}$ such that the matching number (the size of a maximum matching) is $m$. I have been trying to compute the value of ...
28
votes
5answers
2k views

Is the Laplacian on a manifold the limit of graph Laplacians?

Here's the sort of thing I have in mind. Let $M$ be a Riemannian manifold, compact if it helps, and let $\Delta_M$ be the Laplace-Beltrami operator. Choose a sequence of triangulations of $M$ so ...
5
votes
2answers
155 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
2answers
327 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
1answer
182 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 | ...
2
votes
0answers
135 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 ...
0
votes
0answers
65 views

Constructing a digraph from its spectrum

This is related to the following question from cs theory stack exchange: http://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem So it seems as if given a sequence of real ...
-1
votes
1answer
207 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 ...
2
votes
0answers
201 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
1answer
218 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
1answer
280 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 ...
8
votes
0answers
1k views

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 chromatic ...
6
votes
3answers
548 views

Generalization or Improvement of Cheeger inequality on Graphs

Let $G=(V,E)$ be an undirected graph with vertex set $V$ and edge set $E$. Let $A$ denote the adjacency matrix of $G$ and $D$ denote the diagonal matrix such that $D_{i,i}$ equals to the degree $d_i$ ...
1
vote
1answer
182 views

PR[$\lambda_2 > x$] in $G_{np}$ model

Hi! Does anyone know of any statement relating the probability that the second largest eigenvalue of a random graph is bigger than $x$ to the parameter $p$ in the $G_{np}$ model?
2
votes
2answers
254 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 ...
1
vote
2answers
127 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
0answers
223 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
0answers
223 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 ...
11
votes
1answer
252 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, ...
3
votes
1answer
171 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 ...
2
votes
2answers
367 views

Decomposition of $K_{10}$ in copies of the Petersen graph

It is a well-known and cute exercise in algebraic graph theory to show that $K_{10}$ cannot be written as the edge-disjoint union of three copies of the Petersen graph $P$. Indeed, the graph $G$ whose ...
1
vote
0answers
64 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
0answers
118 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
1answer
206 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 ...
4
votes
0answers
123 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 ...
1
vote
2answers
199 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 ...
1
vote
1answer
98 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
1answer
489 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
0answers
87 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
0answers
162 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 ...