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4
votes
1answer
152 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
votes
1answer
189 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 ...
10
votes
0answers
137 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 ...
9
votes
0answers
207 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 ...
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 ...
7
votes
0answers
179 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 ...
5
votes
0answers
134 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 ...
4
votes
0answers
210 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 ...
3
votes
0answers
109 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 ...
3
votes
0answers
207 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 = ...
3
votes
0answers
115 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 ...
3
votes
0answers
106 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 ...
2
votes
0answers
126 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
0answers
183 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 ...
2
votes
0answers
100 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
49 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 ...
1
vote
0answers
63 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
115 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 ...
1
vote
0answers
65 views

Spectra of lifts?

Hi everyone, we know that the lifted(covering) graph inherits every eigenvalue of the base graph. Friedman called eigenvalues of base graph the "old eigenvalues" of its lift. My question is, what can ...
1
vote
0answers
86 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 ...
0
votes
0answers
67 views

How could I prove this eduality 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 ...
0
votes
0answers
59 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 ...
0
votes
0answers
61 views

Spectrum gap of large random weighted semiregular bipartite graph

Hi I need the bound for the spectrum gap of random semiregular ($\ell$, $r$)-bipartite graph. This paper (http://arxiv.org/abs/1212.5216) gives the bound for $\ell$-regular bipartite graphs (with ...
0
votes
0answers
55 views

Proving a lower bound for the maximal eigen-value of a non-negative, irreducible, integer matrix

$A$ is a non-negative, integer, irreducible, $m$ by $m$ matrix. It is well known (Perron-Frobenius) that $A$ has a positive eigen value (denote it by $\lambda$) with a positive eigen vector ($x$). It ...
0
votes
0answers
85 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 ...
0
votes
0answers
85 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 ...
0
votes
0answers
120 views

A graph eigenvalue problem

This is motivated by a graph problem considered by me. For a directed graph $G$ on nodes ${1,\cdots,N}$, denote its graph Laplacian by $L$($l_{ij}=-1$ iff there is an directed edge $j\rightarrow i$ ...