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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Weighted eigenvector-entry sums of bipartite graphs

Let $\lambda_m$ denote the $m$-th eigenvalue of the adjacency matrix $A$ of a bipartite graph $G$ of order $N$ and ${\bf v}_m$ the normalized eigenvector corresponding to $\lambda_m$. I am looking for ...
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2answers
83 views

Matching polynomials and Ramanujan graphs

Is it purely coincidental that the same number $2\sqrt{d-1}$ appears in these two following apparently disparate concepts? A $d-$regular graph is said to be called Ramanujan if its adjacency ...
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37 views

Discrete bi-Laplacian

I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges. In particular ...
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39 views

About adjacency matrices of $k-$shift lifts of graphs

I am finding the notation of cyclic lifts of graphs to be very confusing. Lets say one is looking at a cyclic $k-$lift of a $\vert V \vert$ sized graph. I would like to understand what is the ...
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1answer
91 views

Graph lifts and representation theory

Is there any connection known between the two? One can naturally define lifts of graphs by groups like $\mathbb{Z}_k$ and hence I wonder if representation theoretic properties can be used to say ...
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1answer
87 views

When is a $2$-lift of a graph connected? [closed]

Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ ...
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2answers
160 views

Extremal eigenvalues & eigenvectors of skew-adjacency matrix

I am looking for ways to obtain the extremal eigenvalues and eigenvectors of the skew-adjacency matrix of a directed graph without diagonalizing it. The graphs I am interested in are not regular (but ...
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62 views

Signed Laplacians and Ramanujan graphs

Given a signing/2-lift matrix $A_s$ of a $d-$regular graph one has the relationship that the ``Signed Laplacian" is $L = d + A_s$. This $L$ is still the same size as the base graph. But the lifted ...
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1answer
56 views

Laplacian spectrum of $2-$lifts of graphs

We know that a $2-$ lift of a graph is specified by a $\pm 1$ assignment on the edges of the graph ( given as a signing matrix) denoting which edge is to be duplicated by the identity permutation on ...
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1answer
140 views

How many cospectral graphs available for a given number of nodes?

Two graphs are said to be cospectral if they have same eigenvalues wrt adjacency matrix, Normalised or Signless laplacian matrix. How many graphs has cospectral mates for a given number of nodes? We ...
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131 views

The Bilu-Linial conjecture and Ramanujan graphs

The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...
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1answer
135 views

Are there good ways of relating a minor to the full determinant?

Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
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1answer
66 views

Spectral radius of a time-varying matrix with strictly positive increment of the matrix's entry

Consider a time varying non-negative matrix $A(t)$ and its spectral radius $\rho(A(t))$ being the largest eigenvalue of $A(t)$ and $t$ denotes the time. If $A(t)$ changes over time with each time a ...
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2answers
647 views

What is a “Ramanujan Graph”?

I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify. (1) The Hoory-Linial-Wigderson review on ...
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17 views

Looking to derive bound for modulus of harmonic eigenfunction on weighted graph

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...
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58 views

Is there an elliptic Harnack equality for directed graphs?

The elliptic Harnack inequality for undirected graphs was proven by Delmotte in the paper "Inegalite de Harnack elliptique sur les graphes" (French, ...
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1answer
236 views

Do perfect matching(s) have signatures in the graph eigenvalues?

If the edges of a bipartite graph are such that they can be seen as a disjoint union of perfect matchings then will this somehow reflect in the eigenvalues of the Laplacian? It would be helpful to ...
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2answers
136 views

Graphs whose degree vectors coincide for all powers of their adjacency matrices

Let symmetric $A,B \in \{0, 1\}^{n \times n}$ denote the adjacency matrices of two simple graphs. Further let $\mathbf{1}$ denote the all-one-vector. Now assume that $A^k \mathbf{1} = B^k \mathbf{1}$ ...
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4answers
293 views

Are these three different notions of a graph Laplacian?

I seem to see three different things that are being called the Laplacian of a graph, One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...
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3answers
379 views

Is there a continuous analogue of Ramanujan graphs?

I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its ...
4
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0answers
116 views

minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...
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2answers
105 views

About the roots of the matching polynomial

Can someone kindly give me an expository reference on matching polynomial and its roots? (there is a proof that they are always real?) I saw these two related discussions, Roots of matching ...
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65 views

Are there two-sided $\varepsilon$-expanders with independent sets of size $(1-\varepsilon)n$?

Terry Tao's notes on expander graphs has the following exercise: Exercise 13 Let $G$ be a $k$-regular graph on $n$ vertices that is a two-sided $\epsilon$-expander for some $n > k \geq 1$ and ...
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179 views

A question about Assaf Naor's review in Bourbaki about the Batson-Spielman-Srivastava result

I am referring to this article - http://www.cims.nyu.edu/~naor/homepage%20files/Exp.1033.pdf If I understand right, the author states that his equations (8) and (9) are equivalent to the equations ...
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1answer
156 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) ...
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2answers
110 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 $q\in \mathbb Q$?
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2answers
128 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: $$ ...
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163 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 ...
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1answer
115 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 ...
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2answers
146 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 ...
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1answer
226 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 ...
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1answer
171 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 ...
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1answer
124 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 ...
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1answer
92 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
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2answers
304 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 $G$ ...
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0answers
23 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 ...
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1answer
202 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 ...
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1answer
90 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 ...
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145 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 ...
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0answers
73 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 ...
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3answers
299 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
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1answer
113 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 ...
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1answer
131 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 ...
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2answers
169 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
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2answers
541 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 ...
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1answer
213 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 | ...
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1answer
318 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 ...
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0answers
141 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
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0answers
229 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 ...
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1answer
259 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 ...