TY - JOUR
T1 - A simple proof of an inequality connecting the alternating number of independent sets and the decycling number
AU - Levit, Vadim E.
AU - Mandrescu, Eugen
PY - 2011/7/6
Y1 - 2011/7/6
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
UR - http://www.scopus.com/inward/record.url?scp=79955474454&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2010.06.004
DO - 10.1016/j.disc.2010.06.004
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AN - SCOPUS:79955474454
SN - 0012-365X
VL - 311
SP - 1204
EP - 1206
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 13
ER -