A note on the alternating number of independent sets in a graph
Document Type
Article
Publication Date
9-2026
Journal / Book Title
Discrete Mathematics
Abstract
The independence polynomial of a graph G evaluated at −1, denoted here as , has arisen in a variety of different areas of mathematics and theoretical physics as an object of interest. Engström used discrete Morse theory to prove that where is the decycling number of G, i.e., the minimum number of vertices needed to be deleted from G so that the remaining graph is acyclic. Here, we improve Engström's bound by showing where is the minimum number of vertices needed to be deleted from G so that the resulting graph contains no induced cycles whose length is divisible by 3. We also note that this bound is not just sharp but that every value in the range given by the bound is attainable by some connected graph.
DOI
10.1016/j.disc.2026.115147
MSU Digital Commons Citation
Cutler, Jonathan; Kahl, Nathan; and Zielonka, Phoebe Rose, "A note on the alternating number of independent sets in a graph" (2026). Department of Mathematics Facuty Scholarship and Creative Works. 206.
https://digitalcommons.montclair.edu/mathsci-facpubs/206
Published Citation
Cutler, Jonathan, et al. “A Note on the Alternating Number of Independent Sets in a Graph.” Discrete Mathematics, vol. 349, no. 9, Sept. 2026, p. 115147. https://doi.org/10.1016/j.disc.2026.115147.