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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2
votes
2answers
677 views

Positive semidefinite decomposition, Laplacian eigenvalues, and the oriented incidence matrix

Suppose $A\in\mathbb{C}^{n\times n}$ is Hermitian and positive semidefinite with some decomposition $A=BB^*$, where $B=(b_{ij})\in\mathbb{C}^{n\times m}$ (not necessarily the Cholesky decomposition). ...
7
votes
1answer
452 views

Diophantine elements in SU(2)

Following notions from [1], call a set of elements $g_1, \dots, g_k \in G = SU(2)$ Diophantine if it satisfies the following property: there exists a constant $D$ such that for every word $W_m$ of ...
2
votes
2answers
466 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 ...
2
votes
2answers
379 views

Isoperimetric dimension of Graphs.

According to the wikipedia page on "Isoperimetric dimension", the isoperimetric dimension is invariant under quasi-isometries, even between manifolds and graphs: "[...] the isoperimetric dimension is ...
11
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 ...
3
votes
1answer
187 views

Estimation of DS graph growth

We know that $DS$ graphs are such connected graphs that determinable by their adjacency spectrum. Suppose $DS(n)$ and $G(n)$ show the number of $DS$ graphs and all graphs with $n$ vertices,...
2
votes
3answers
531 views

Non-isomorphic graphs with the same numbers of closed walks

Can somebody help me to construct two family of finite simple connected graph $G_i$ and $H_i$, $i=1, 2, \cdots,n$ ($n$ possibly large), such that: $1)$ $G_i‎\ncong H_i$ for $i=1, 2, \cdots, n$ $2)$ $...
5
votes
3answers
311 views

Operation on Isospectral graphs

Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that are obtained by applying a binary operation to $G$ and $H$? For example, to take one ...
7
votes
2answers
414 views

cospectral graphs

The simple connected graph $G$ has $n$ vertices and we have: 1) $|E(G)|‎\geq‎ \frac{n(n-1)}{3}$ 2) we have the spectrum and degree sequence of $G$ 3) $Spectrum(G)=Spectrum(H)$ Is $G \cong ‎H$?
1
vote
1answer
272 views

spectrum and degree sequence

We have the spectrum and the degree sequence of one graph. Can we uniquely determine the graph with these given information?
1
vote
2answers
362 views

Lovasz theta function of circulant graphs

Let $G$ be a cirulant graph with no loops at vertices and vertex degree $d$. Is the Lovasz theta function of this graph given by: $\vartheta(G) = \max_{i}\frac{-N\epsilon_{i}}{-\epsilon_{i}+d-1}$? ...
0
votes
1answer
257 views

Lovasz theta function - uses

Lovasz theta function bounds the Shannon capacity of graphs. What are some other uses of the function - especially in asymptotic coding theory and optimization problems?
0
votes
1answer
108 views

Counting walks on proper colorings of odd cycles

Let $G$ be an undirected odd cycle. Let $f$ be a proper 3-coloring of $G$. If $w=v_1v_2...v_k$ is a walk on $k$ vertices of $G$, let $f(w)=f(v_1)f(v_2)...f(v_k)$. Let $W_k=\{f(w)|w$ is a walk on $k$...
15
votes
2answers
851 views

Groups with a rational generating function for the word problem

This question comes more from curiosity than a specific research problem. Let G be a group and S a finite symmetric generating set. By the WP(G,S) I mean the set of all words in the free monoid on S ...
5
votes
3answers
489 views

Bounds on maximal eigenvalue of a k-regular graph

Given a k-regular graph $G$ (every vertex is of degree k), one defines its Laplace operator as $L(G)=D-A=kI-A$, where $I$ is identity matrix and $A$ adjacency matrix of $G$. Let $\lambda_{1}\leq \...
3
votes
2answers
715 views

Effect of different graph operations on spectrum of graph laplacian?

The algebraic connectivity of a graph G is the second-smallest eigenvalue of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected graph. The magnitude of this ...
7
votes
3answers
2k views

Random bipartite graphs

Consider the following situation: I have a set $A$ of $n$ vertices and a set $B$ of $N = n^2$vertices. I consider the bipartite graph $(A, B)$ and put at random $M = n^{1 + \varepsilon}$ edges (or I ...
10
votes
3answers
683 views

Eigenfunctions of random graphs

Consider a random $d$-regular graph on $n$ vertices. What can be said about its nontrivial (i.e. orthogonal to the constant) eigenfunctions? For example, I'm interested whether there are "nodal zones",...
5
votes
1answer
304 views

Second eigenvalue of suspension of a graph

Suppose I have some $d$-regular graph $G$. Let $\lambda = \max\{\lambda_2(G), |\lambda_n(G)|\}$ be the second largest eigenvalue of the adjacency matrix of $G$. Now take $\tilde{G}$, the suspension of ...
3
votes
2answers
1k views

Complex Eigenvalues of Directed Graphs

I have been computing eigenvalues of adjacency matrices for several directed (not necessarily strongly connected) graphs and one remarkable property seemed to hold (each graph that I have examined ...
4
votes
3answers
867 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 ...
33
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 ...
3
votes
4answers
785 views

Spectral gap for random bipartite regular graphs

For a graph $G$, let its Laplacian be $\Delta = I - D^{-1/2}AD^{-1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm interested ...
15
votes
8answers
6k views

The first eigenvalue of a graph - what does it reflect?

A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the ...
1
vote
1answer
188 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?
6
votes
3answers
651 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 $d_i$ ...
8
votes
3answers
510 views

Classes of graphs for which isospectrum implies isomorphism ?

The spectrum of a graph is the (multi)set of eigenvalues of its adjacency matrix (or Laplacian, depending on what you're interested in). In general, two non-isomorphic graphs might have the same ...
8
votes
2answers
765 views

Spectrum of the Laplacian on G(n, p) and G(n, M)

A random graph in $G(n, p)$ model is a graph on $n$ vertices in which for each of the $n\choose{2}$ edges we independently flip a coin, then take the edge with probability $p$ or remove it with $1 - p$...
8
votes
1answer
737 views

When the Lovász theta-function saturates its upper bound

The Lovász $\vartheta$-function of a graph $G$, $\vartheta(G)$, is well-known to be "sandwiched" between the independence number of the graph, $\alpha(G)$, and the chromatic number of its complement, $...
2
votes
4answers
982 views

On the spectrum of random regular graph

For a random $d$-regular graph, where $d$ can be fixed or can grow slowly with the size of the graph $n$, what can we say about its spectrum - Do you believe it has simple spectrum? Thank you,