The Number of Independent Sets in a Regular Graph
نویسندگان
چکیده
منابع مشابه
The Number of Independent Sets in a Regular Graph
We show that the number of independent sets in an N -vertex, d-regular graph is at most (2−1), where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also ...
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Let Gm,n be the m×n grid graph. That is, the vertices of Gm,n are the (m+1)(n+1) points (i, j) (0 ≤ i ≤ m, 0 ≤ j ≤ n) in the plane, and its edges are all of the pairs (i, j), (i′, j′) of vertices for which |i − i′| + |j − j′| = 1. Let f(m,n) be the number of independent sets of vertices in Gm,n. We study the growth of f(m,n). The figure below shows an independent set S in G4,6. figure=grid.eps ...
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This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. The main motivating problem is the following: in the family of d-regular graphs, which graph G maximize/minimize the quantity i(G), the number of independent sets in G normalized exponentially by the size of G? What if i(G) is replaced by some other grap...
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ژورنال
عنوان ژورنال: Combinatorics, Probability and Computing
سال: 2009
ISSN: 0963-5483,1469-2163
DOI: 10.1017/s0963548309990538