A simple proof of an inequality connecting the alternating number of independent sets and the decycling number

Vadim E. Levit; Eugen Mandrescu

doi:10.1016/j.disc.2010.06.004

A simple proof of an inequality connecting the alternating number of independent sets and the decycling number

Vadim E. Levit
, Eugen Mandrescu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

If ^sk denotes the number of independent sets of cardinality k and α(G) is the size of a maximum independent set in graph G, then I(G;x)=^s0+^s1x+⋯+sα( _G)xα(^G) is the independence polynomial of G (Gutman and Harary, 1983) [8]. In this paper we provide an elementary proof of the inequality|I(G;-1)|≤2^φ(G) (Engstrm, 2009) [7], where φ(G) is the decycling number of G (Beineke and Vandell, 1997) [3], namely, the minimum number of vertices that have to be deleted in order to turn G into a forest.

Original language	English
Pages (from-to)	1204-1206
Number of pages	3
Journal	Discrete Mathematics
Volume	311
Issue number	13
DOIs	https://doi.org/10.1016/j.disc.2010.06.004
State	Published - 6 Jul 2011

Keywords

Cyclomatic number
Decycling number
Forest
Independence polynomial
Independent set

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10.1016/j.disc.2010.06.004

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abstract = "If sk denotes the number of independent sets of cardinality k and α(G) is the size of a maximum independent set in graph G, then I(G;x)=s0+s1x+⋯+sα( G)xα(G) is the independence polynomial of G (Gutman and Harary, 1983) [8]. In this paper we provide an elementary proof of the inequality|I(G;-1)|≤2φ(G) (Engstrm, 2009) [7], where φ(G) is the decycling number of G (Beineke and Vandell, 1997) [3], namely, the minimum number of vertices that have to be deleted in order to turn G into a forest.",

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AU - Mandrescu, Eugen

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N2 - If sk denotes the number of independent sets of cardinality k and α(G) is the size of a maximum independent set in graph G, then I(G;x)=s0+s1x+⋯+sα( G)xα(G) is the independence polynomial of G (Gutman and Harary, 1983) [8]. In this paper we provide an elementary proof of the inequality|I(G;-1)|≤2φ(G) (Engstrm, 2009) [7], where φ(G) is the decycling number of G (Beineke and Vandell, 1997) [3], namely, the minimum number of vertices that have to be deleted in order to turn G into a forest.

AB - If sk denotes the number of independent sets of cardinality k and α(G) is the size of a maximum independent set in graph G, then I(G;x)=s0+s1x+⋯+sα( G)xα(G) is the independence polynomial of G (Gutman and Harary, 1983) [8]. In this paper we provide an elementary proof of the inequality|I(G;-1)|≤2φ(G) (Engstrm, 2009) [7], where φ(G) is the decycling number of G (Beineke and Vandell, 1997) [3], namely, the minimum number of vertices that have to be deleted in order to turn G into a forest.

KW - Cyclomatic number

KW - Decycling number

KW - Forest

KW - Independence polynomial

KW - Independent set

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A simple proof of an inequality connecting the alternating number of independent sets and the decycling number

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Keywords

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