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# On Symmetry of Independence Polynomials

^{1}Department of Computer Science and Mathematics, Ariel University Center of Samaria, Kiryat HaMada, Ariel 40700, Israel

^{2}Department of Computer Science, Holon Institute of Technology, 52 Golomb Street, Holon 58102, Israel

* Author to whom correspondence should be addressed.

Received: 27 April 2011 / Revised: 20 June 2011 / Accepted: 22 June 2011 / Published: 15 July 2011

(This article belongs to the Special Issue Symmetry Measures on Complex Networks)

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# Abstract

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum matching. If sk is the number of independent sets of size k in G, then I(G; x) = s_{0}+ s

_{1}x + s

_{2}x

_{2}+ ... + s

_{α}x

^{α}, α = α (G), is called the independence polynomial of G (Gutman and Harary, 1986). If sj = s

_{α}j for all 0 ≤ j ≤ [α/2], then I(G; x) is called symmetric (or palindromic). It is known that the graph G

_{°}2K

_{1}, obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanović, 1998). In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to show that for every graph G and for each non-negative integer k ≤ μ (G), one can build a graph H, such that: G is a subgraph of H, I (H; x) is symmetric, and I (G

_{° }2K1; x) = (1 + x)

^{k}· I (H; x).

*This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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