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On Symmetry of Independence Polynomials
Department of Computer Science and Mathematics, Ariel University Center of Samaria, Kiryat HaMada, Ariel 40700, Israel
Department of Computer Science, Holon Institute of Technology, 52 Golomb Street, Holon 58102, Israel
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Received: 27 April 2011; in revised form: 20 June 2011 / Accepted: 22 June 2011 / Published: 15 July 2011
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) = s0 + s1x + s2x2 + ... + sαxα, α = α (G), is called the independence polynomial of G (Gutman and Harary, 1986). If sj = sαj for all 0 ≤ j ≤ [α/2], then I(G; x) is called symmetric (or palindromic). It is known that the graph G ° 2K1, 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).
Keywords: independent set; independence polynomial; symmetric polynomial; palindromic polynomial
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MDPI and ACS Style
Levit, V.E.; Mandrescu, E. On Symmetry of Independence Polynomials. Symmetry 2011, 3, 472-486.
Levit VE, Mandrescu E. On Symmetry of Independence Polynomials. Symmetry. 2011; 3(3):472-486.
Levit, Vadim E.; Mandrescu, Eugen. 2011. "On Symmetry of Independence Polynomials." Symmetry 3, no. 3: 472-486.