Eugen Mandrescu and Alexander Spivak

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 22, 2016, Number 2, Pages 44—53

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## Details

### Authors and affiliations

Eugen Mandrescu

*Department of Computer Sciences, Holon Institute of Technology
52 Golomb Str., Holon 5810201, Israel
*

Alexander Spivak

*Department of Computer Sciences, Holon Institute of Technology*

52 Golomb Str., Holon 5810201, Israel

52 Golomb Str., Holon 5810201, Israel

### Abstract

If *s _{k}* denotes the number of independent sets of cardinality

*k*in graph

*G*, and

*α*(

*G*) is the size of a maximum independent set, then

is the independence polynomial of

*G*(I. Gutman and F. Harary, 1983, [8]). The Merrifield–Simmons index

*σ*(

*G*) (known also as the Fibonacci number) of a graph

*G*is defined as the number of all independent sets of

*G*. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdos (1987, [2]) conjectured that

*I*(

*T*,

*x*) is unimodal whenever

*T*is a tree, while, in general, they proved that for each permutation

*π*of {1, 2, …,

*α*} there is a graph

*G*with

*α*(

*G*) =

*α*such that

*s*

_{π(1)}<

*s*

_{π(2)}< … <

*s*

_{π(α)}. By maximal tree on

*n*vertices we mean a tree having a maximum number of maximal independent sets among all the trees of order

*n*. B. Sagan proved that there are three types of maximal trees, which he called batons [24].

In this paper we derive closed formulas for the independence polynomials and Merrifield–Simmons indices of all the batons. In addition, we prove that *I*(*T*,*x*) is log-concave for every maximal tree *T* having an odd number of vertices. Our findings give support to the above mentioned conjecture.

### Keywords

- Independent set
- Independence polynomial
- Log-concave sequence
- Tree

### AMS Classification

- 05C05
- 05C69
- 05C31

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## Cite this paper

APAMandrescu, E., & Spivak, A. (2016). Maximal trees with log-concave independence polynomials. Notes on Number Theory and Discrete Mathematics, 22(2), 44-53.

ChicagoMandrescu, Eugen, and Alexander Spivak. “Maximal Trees with Log-concave Independence Polynomials.” Notes on Number Theory and Discrete Mathematics 22, no. 2 (2016): 44-53.

MLAMandrescu, Eugen, and Alexander Spivak. “Maximal Trees with Log-concave Independence Polynomials.” Notes on Number Theory and Discrete Mathematics 22.2 (2016): 44-53. Print.