Roman Bacik, Sanjeev Mahajan

The graph homomorphism problem is a canonical $NP$-complete problem.

It generalizes various other well-studied problems such as graph

coloring and finding cliques. To get a better insight into a

combinatorial problem, one often studies relaxations of the problem.

We define fractional homomorphisms and pseudo-homomorphisms ...
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Meena Mahajan, V Vinay

In this paper we approach the problem of computing the characteristic

polynomial of a matrix from the combinatorial viewpoint. We present

several combinatorial characterizations of the coefficients of the

characteristic polynomial, in terms of walks and closed walks of

different kinds in the underlying graph. We develop algorithms based

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Oded Goldreich, Dana Ron

We consider testing graph expansion in the bounded-degree graph model.

Specifically, we refer to algorithms for testing whether the graph

has a second eigenvalue bounded above by a given threshold

or is far from any graph with such (or related) property.

We present a natural algorithm aimed ... more >>>

Eldar Fischer

Let $P$ be a property of graphs. An $\epsilon$-test for $P$ is a

randomized algorithm which, given the ability to make queries whether

a desired pair of vertices of an input graph $G$ with $n$ vertices are

adjacent or not, distinguishes, with high probability, between the

case of $G$ satisfying ...
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Noga Alon, Klim Efremenko, Benny Sudakov

Let $G=(V,E)$ be a connected undirected graph with $k$ vertices. Suppose

that on each vertex of the graph there is a player having an $n$-bit

string. Each player is allowed to communicate with its neighbors according

to an agreed communication protocol, and the players must decide,

deterministically, if their inputs ...
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