# The Number of Independent Sets in a Graph with Small Maximum Degree

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## Abstract

Let ind( (where with equality if and only if

*G*) be the number of independent sets in a graph*G*. We show that if*G*has maximum degree at most 5 then$${\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}$$

*d*(·) is vertex degree, iso(*G*) is the number of isolated vertices in*G*and*K*_{a,b}is the complete bipartite graph with*a*vertices in one partition class and*b*in the other), with equality if and only if each connected component of*G*is either a complete bipartite graph or a single vertex. This bound (for all*G*) was conjectured by Kahn. A corollary of our result is that if*G*is*d*-regular with 1 ≤*d*≤ 5 then$${\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d},$$

*G*is a disjoint union of |*V*(*G*)|/2*d*copies of*K*_{d,d}. This bound (for all*d*) was conjectured by Alon and Kahn and recently proved for all*d*by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most 3 the search could be done by hand, but for the case of maximum degree 4 or 5, a computer is needed.### Keywords

Independent set Stable set Regular graph### Mathematics Subject Classification (2000)

Primary 05C69 Secondary 05A16## Preview

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### References

- 1.Alon N.: Independent sets in regular graphs and sum-free subsets of finite groups. Israel J. Math.
**73**, 247–256 (1991)MATHCrossRefMathSciNetGoogle Scholar - 2.Kahn J.: An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput.
**10**, 219–237 (2001)MATHMathSciNetGoogle Scholar - 3.Zhao Y.: The number of independent sets in a regular graph. Combin. Probab. Comput.
**19**, 315–320 (2010)MATHCrossRefMathSciNetGoogle Scholar

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