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3 votes
0 answers
61 views

Is this bipartite equivalent of 1-walk-regular graphs known?

A graph $G$ is 1-walk-regular if for each vertex $v$ the number of closed walks of length $\ell$ starting (and ending) at $v$ depends only on $\ell$ but not on $v$. for each edge $vw$ the number of ...
M. Winter's user avatar
  • 13.6k
1 vote
1 answer
177 views

Spectral characterization of complete or complete bipartite graphs

The Lemma 6 in this paper mention the following spectral characterization of complete or complete bipartite graphs: Let $G$ be a connected graph with $\ge 2$ vertices. Then $\lambda_2=...=\lambda_{n-...
YuiTo Cheng's user avatar
0 votes
0 answers
55 views

Comparing spectral radius of two graphs using the entry of Perron vector

Suppose we have a graph $G$. Let $A$ be the adjacency matrix of $G$ and $x$ be the corresponding Perron vector. Let $x = (x_1,x_2,\cdots,x_n)^t$, where $x_i$ corresponds to the vertex $i \in V(G)$. We ...
User8976's user avatar
  • 199
0 votes
0 answers
120 views

Does an extension of the B.E.S.T. theorem for multiple Eulerian circuits exist?

Given a directed multigraph $G=(V,E)$ (multiple edges and loops are permitted) the number of distinct Eulerian circuits for $G$ can be calculated with the B.E.S.T. theorem. Does a similar theory for ...
Ben Deitmar's user avatar
  • 1,295
4 votes
0 answers
176 views

Can resolution of the Kadison-Singer Problem provide progress on the Komlos Conjecture?

This is not a concrete question, just some thoughts. The Komlos Conjecture is as follows- There exists an absolute constant $C>0$, such that the following holds: For all $d$ and any set of vectors ...
Sudipta Roy's user avatar
6 votes
1 answer
744 views

Relationship between spectral gaps of adjacency and Laplacian matrices of graphs

Let $G$ be an undirected simple graph on $n$ vertices, with self-loops allowed, and with arbitrary positive edge weights $w_{u,v}$ (which is $0$ if there is no edge between $u$ and $v$). Let $A$ be ...
Vilas Winstein's user avatar
3 votes
1 answer
143 views

Reference request: Spectrum of intersection matrices

Let $P(A)$ be the set of all non-empty proper subsets of a finite set $A$. Let $M$ be a matrix indexed by the set in $P(A)$ whose $ij$ the entry is $1$ if the associated sets are disjoint and $0$ ...
GA316's user avatar
  • 1,269
34 votes
1 answer
789 views

Which graphs on $n$ vertices have the largest determinant?

This is a question that seems like it should have been studied before, but for some reason I cannot find much at all about it, and so I am asking for any pointers / references etc. The determinant of ...
Gordon Royle's user avatar
  • 12.7k
1 vote
0 answers
121 views

Cheeger constant of truncated hypercube

Look at the $d$-dimensional hypercube and truncate it. This means one replaces each vertex by a cycle (of length $d$) in such a way the the new graph is 3-regular. Question 1: What is the asymptotic ...
ARG's user avatar
  • 4,422
9 votes
1 answer
261 views

Expansion in strongly regular graphs

Have you seen the following statement proven anywhere? Let $G$ be a strongly regular graph with parameters $(n,k,\lambda,\mu)$ with $\lambda,\mu>0$. Then there is no set $A$ of at least $n/4$ ...
EJI's user avatar
  • 93
17 votes
8 answers
10k 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 ...
Seva's user avatar
  • 23k