Independent set and matching permutations

Ball, Taylor; Galvin, David; Hyry, Catherine; Weingartner, Kyle

doi:10.1002/jgt.22724

Cited by 1 publication

(3 citation statements)

References 14 publications

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“…As a part of the computations, in order to support the Alavi, Malde, Schwenk, and Erdös conjecture [19], using the linear algorithm described above, it was verified that for all trees up to 25 vertices, their independence polynomials are log-concave (and, consequently, unimodal), see also ref. [21]. All of the sudden, when the number of vertices of a tree reached 26, there were found two trees having their independence polynomials unimodal but not log-concave.…”

Section: Applications Of the Algorithmmentioning

confidence: 98%

“…ðÞ , is the independence polynomial of G [1], the independent set polynomial of G [2], or the stable set polynomial of G [3]. Some updated observations concerning the independence polynomial may be found in [4,5].…”

mentioning

confidence: 99%

“…To this end, it supports the unimodality and log-concavity of independence polynomials of trees with up to 20 vertices. Further, Radcliffe has verified that independence polynomials of trees up to 25 vertices are log-concave [21].…”

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confidence: 99%

See 2 more Smart Citations

On Computing of Independence Polynomials of Trees

Ohr¹,

Levit²,

Ron³

et al. 2023

Recent Research in Polynomials [Working Title]

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An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote the cardinality of a maximum independent set in the graph $G = (V, E)$. Gutman and Harary defined the independence polynomial of $G$ \[ I(G;x) = \sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)}, \] where $s_k$ denotes the number of independent sets of cardinality $k$ in the graph $G$. A comprehensive survey on the subject is due to Levit and Mandrescu, where some recursive formulas are allowing computation of the independence polynomial. A direct implementation of these recursions does not bring about an efficient algorithm. In 2021, Yosef, Mizrachi, and Kadrawi developed an efficient way for computing the independence polynomials of trees with $n$ vertices, such that a database containing all of the independence polynomials of all the trees with up to $n-1$ vertices is required. This approach is not suitable for big trees, as an extensive database is needed. On the other hand, using dynamic programming, it is possible to develop an efficient algorithm that prevents repeated calculations. In summary, our dynamic programming algorithm runs over a tree in linear time and does not depend on a database.

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Section: Applications Of the Algorithmmentioning

confidence: 98%

mentioning

confidence: 99%