# Questions tagged [spectral-graph-theory]

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### Complexity to compute the eigenvalue signs of the adjacency matrix

Let $A$ be the $n\times n$ adjacency matrix of a (non-bipartite) graph. Assume that we are given the amplitudes of its eigenvalues, i.e., $|\lambda_1|=a_1,\ldots, |\lambda_n|=a_n$, and we would like ...
158 views

### Generating a random graph with constraints on spectrum

Consider two sequences $u_1 \geq u_2 \geq ... \geq u_n$ and $l_1 \geq l_2 \geq ... \geq l_n$ with $u_i \geq l_i$ for every $i$. Let $\mathcal{G}(l_{1:n},u_{1:n})$ be all undirected unweighted simple ...
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### Spectral Graph Theory and Matroid Theory

I have just started grad school this year and I have been into Spectral Graph Theory for some time now. Recently I got introduced to Matroid Theory and although I know the field has been around for ...
1k views

### Eigenvalues of adjacency matrix of a connected bipartite graph

Let $G=(V,E)$ is a connected d-regular bipartite graph with the same number of vertices on both sides of the bipartition. It's known that that the largest eigenvalue of its adjacency matrix would be d,...
58 views

### Can we always find a graph with a given algebraic connectivity?

This is crossposted from math stackexchange. This is my first time posting here, so let me know if I'm doing something wrong. I would like to experiment with various spectral properties of graphs, ...
75 views

### The algebraic connectivity of graphs with large isoperimetric number

I asked this question on MO, but no answer. Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$. the isoperimetric number of $G$, denoted $i(G)$, is defined by i(G) = \min_{|S| \leq |...
155 views

### Construction of a graph which has regular subgraphs at each iteration of a recursive process

I am studying Graph Isomorphism and also trying to figure out the complexity of a certain class of graph. The graph I am studying at the moment is described below Description : $G$ is a $r$ regular ...
204 views

### Graph sparsification

I would like to ask if any one is aware of any results related to graph sparsification with bounded degrees? What I mean is anyone aware of graph sparsification results such that the resultant graph ...
156 views

### Graph has several MST what does it mean combinatorically?

This question is not theoretical, it's about combinatorial meaning. In graph theory there is a notion of complexity of a graph, which is equal to the number of spanning trees in a graph, which ...
91 views

### Purely spectral algorithms for the minimum spanning tree problem

There are many algorithms that address the MST problem, from classical (Boruvka, Prim, Kruskal), optimal (Pettie et al.) to their distributed variants (Bader et al.). However, I fail to find any ...
108 views

### Spectra of Cayley Graphs for non-Abelian Groups

There appears to be much interest in the subject of the spectra of Cayley graphs. Indeed it appears that the spectra are highly related or may be computed through the irreducible representations of ...
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### Sparse-cut approximation for well connected graphs

Theorem: For any graph $G=(V,E)$, there is a polynomial time algorithm that finds a cut $S \subseteq V$ with conductance at most $\sqrt{2\big(1 − \lambda(G)\big)}$. If I understand UGC correctly, ...
133 views

Can someone kindly help clarify as to exactly what is the generally accepted definition of the "sparsest cut" problem for a graph? (Isn't the set which achieves the Cheeger constant for a ...
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### Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
136 views

### Constructing a digraph from its spectrum

This is related to the following question which has already been explored: Reverse Graph Spectra Problem? So it seems as if given a sequence of real numbers, it is not always possible to generate a 0-...
102 views

### Is there a diagonal matrix D such that DMD is SDD, where M is SPD matrix

Let $M$ be symmetric and positive definite matrix (SPD). It is known  that if $M$ is SPD and in addition satisfies $M_{ij}\leq 0$, for $i\neq j$ (called M-matrix) then there is a positive ...
I am reading the paper above by Lee, Gharan and Trevisan, and I am having trouble with lemma 4.8 on page 23. How do we form the set $T_1, \ldots, T_{r'}$ such that $r'>(1-\delta/2)k$ and each set \$...