On Symmetry of Independence Polynomials

An independent set in a graph is a set of pairwise non-adjacent vertices, and alpha(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while mu(G) is the cardinality of a maximum matching. If s_{k} is the number of independent sets of cardinality k in G, then I(G;x)=s_{0}+s_{1}x+s_{2}x^{2}+...+s_{\alpha(G)}x^{\alpha(G)} is called the independence polynomial of G (Gutman and Harary, 1983). If $s_{j}=s_{\alpha-j}$, 0=<j =<alpha(G), 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 (Stevanovic, 1998). In this paper we show that for every graph G and for each non-negative integer k =<mu(G), one can build a graph H, such that: G is a subgraph of H, I(H;x) is symmetric, and I(G*2K_{1};x)=(1+x)^{k}*I(H;x).


Introduction
Throughout this paper G = (V, E) is a simple (i.e., a finite, undirected, loopless and without multiple edges) graph with vertex set V = V (G) and edge set E = E(G). If X ⊂ V , then G[X] is the subgraph of G spanned by X. By G−W we mean the subgraph G[V − W ], if W ⊂ V (G). We also denote by G − F the partial subgraph of G obtained by deleting the edges of F , for F ⊂ E(G), and we write shortly G− e, whenever F = {e}. The neighborhood of a vertex v ∈ V is the set N G (v) = {w : w ∈ V and vw ∈ E}, and N G [v] = N G (v) ∪ {v}; if there is no ambiguity on G, we write N (v) and N [v]. K n , P n , C n denote, respectively, the complete graph on n ≥ 1 vertices, the chordless path on n ≥ 1 vertices, and the chordless cycle on n ≥ 3 vertices.
The disjoint union of the graphs G 1 , G 2 is the graph G = G 1 ∪G 2 having as vertex set the disjoint union of V (G 1 ), V (G 2 ), and as edge set the disjoint union of E(G 1 ), E(G 2 ). In particular, nG denotes the disjoint union of n > 1 copies of the graph G.
The corona of the graphs G and H with respect to A ⊆ V (G) is the graph (G, A) • H obtained from G and |A| copies of H, such that each vertex of A is joined to all vertices of a copy of H. If A = V (G) we use G • H instead of (G, V (G)) • H (see Figure 1 for an example). Let G, H be two graphs and C be a cycle on q vertices of G. By (G, C) △ H we mean the graph obtained from G and q copies of H, such that each two consecutive vertices on C are joined to all vertices of a copy of H (see Figure 2 for an example). An independent (or a stable) set in G is a set of pairwise non-adjacent vertices. By Ind(G) we mean the family of all independent sets of G. An independent set of maximum size will be referred to as a maximum independent set of G, and the independence number of G, denoted by α(G), is the cardinality of a maximum independent set in G.
Let s k be the number of independent sets of size k in a graph G. The polynomial is called the independence polynomial of G [7], the independent set polynomial of G [11]. In [6], the dependence polynomial D(G; x) of a graph G is defined as D(G; x) = I(G; −x).
A matching is a set of non-incident edges of a graph G, while µ(G) is the cardinality of a maximum matching. Let m k be the number of matchings of size k in G. The polynomial is called the matching polynomial of G [5].
The independence polynomial has been defined as a generalization of the matching polynomial, because the matching polynomial of a graph G and the independence polynomial of its line graph are identical. Recall that given a graph G, its line graph L(G) is the graph whose vertex set is the edge set of G, and two vertices are adjacent if they share an end in G. For instance, the graphs G 1 and G 2 depicted in Figure 3 satisfy G 2 = L(G 1 ) and, hence, In [7] a number of general properties of the independence polynomial of a graph are presented. As examples, we mention that: The following equalities are very useful in calculating of the independence polynomial for various families of graphs. A finite sequence of real numbers (a 0 , a 1 , a 2 , ..., a n ) is said to be: • unimodal if there is some k ∈ {0, 1, ..., n}, such that a 0 ≤ ... ≤ a k−1 ≤ a k ≥ a k+1 ≥ ... ≥ a n ; • log-concave if a 2 i ≥ a i−1 · a i+1 for i ∈ {1, 2, ..., n − 1}. • symmetric (or palindromic) if a i = a n−i , i = 0, 1, ..., ⌊n/2⌋.
It is easy to see that if α(G) ≤ 3 and I(G; x) is symmetric, then it is also log-concave.
In this paper we show that every graph H derived from the graph G by Stevanović's rules [23] gives rise to the following decomposition for every non-negative integer k ≤ µ (G).

Preliminaries
The symmetry of the matching polynomial and the characteristic polynomial of a graph were examined in [13], while for the independence polynomial we quote [10], [23], and [3]. Recall from [13] that G is called a equible graph if G = H • K 1 for some graph H. Both matching polynomials and characteristic polynomials of equible graphs are symmetric [13]. Nevertheless, there are non-equible graphs whose matching polynomials and characteristic polynomials are symmetric. It is worth mentioning that one can produce graphs with symmetric independence polynomials in different ways. We summarize some of them in the sequel.

Gutman's construction [8]
For integers p > 1, q > 1, let J p,q be the graph built in the following manner [8]. Start with three complete graphs K 1 , K p and K q whose vertex sets are disjoint. Connect the vertex of K 1 with p − 1 vertices of K p and with q − 1 vertices of K q . The graph thus obtained has a unique maximum independent set of size three, and its independence polynomial is equal to Hence the independence polynomial of G = J p,q + K pq−p−q+1 is which is clearly symmetric and log-concave.

Bahls and Salazar's construction [3]
Such a graph consists of k copies of K t , each glued to the previous one by identifying certain prescribed subgraphs isomorphic to . Given G and U and a graph H, we write H + (G, U ) instead of (H, V (H)) + (G, U ).
Theorem 2.1 [3] Let t ≥ 2, k ≥ 1, and d ≥ 0 be integers, and let G = (V, E) be a graph with U ⊆ V a distinguished subset of vertices. Suppose that each of the graphs G, G−U , and (G, U )+K 1 have symmetric and unimodal independence polynomials, and that deg( Then the independence polynomial of the graph P (t, k, d) + (G, U ) is symmetric and unimodal.
• A cycle cover of a graph G is a spanning graph of G, each connected component of which is a vertex (which we call a vertex-cycle), an edge (which we call an edge-cycle), or a proper cycle. Let Γ be a cycle cover of G.
Let H n , n ≥ 1, be the graphs obtained according to Rule 3 from P n , as one can see in Figure 8.   Theorem 2.7 [19] If J n (x) = I(H n ; x), n ≥ 0, then (i) J 0 (x) = 1, J 1 (x) = 1 + 3x + x 2 and J n , n ≥ 2, satisfies the following recursive relations: (ii) J n is both symmetric and unimodal.
It was conjectured in [19] that I(H n ; x) is log-concave and has only real roots. This conjecture has been resolved as follows.
Theorem 2.8 [24] Let n ≥ 1. Then (i) the independence polynomial of H n is (ii) I(H n ; x) has only real zeros, and, therefore, it is log-concave and unimodal.

Results
The following lemma goes from the well-known fact that the polynomial P (x) is symmetric if and only if it equals its reciprocal, i.e., and h (x) be polynomials satisfying f (x) = g (x) · h (x). If any two of them are symmetric, then the third is symmetric as well.
For H = 2K 1 , Theorem 1.1 gives 2 and deg (I (G • 2K 1 ; x)) = 2n, one can easily see that the polynomial I (G • 2K 1 ; x) satisfies the identity (*). Thus we conclude with the following.  For S ∈ Ind(G), let denote the following families of independent sets:

Clique covers
Since A is a clique, it follows that |S ∩ A| ≤ 1.
In this case S ∪ W ∈ Ω G1 S if and only if S ∪ W ∈ Ω G2 S . Hence, for each size m ≥ |S|, we get that S has W ∩ V (H) = ∅ for exactly one H, namely, the graph H whose vertices are joined to a. Hence, W may contain vertices only from (|A| − 1) H.
On the other hand, each S ∪W ∈ Ω G2 S has W ∩V (H) = ∅ for the unique H appearing in (G, A) + H. Therefore, W may contain vertices only from (|A| − 1) H.
Hence, for each positive integer m ≥ |S|, we obtain that Consequently, one may infer that for each size, the two graphs, G 1 and G 2 , have the same number of independent sets, in other words, I(G 1 ; x) = I(G 2 ; x).
Since  Theorem 3.5 If G is a graph of order n and Φ is a clique cover, then Proof. Let Φ = {A 1 , A 2 , ..., A q }. According to Corollary 3.4, each (a) vertex-clique of Φ yields (1 + x) 2−2 = 1 as a factor of I(G • 2K 1 ; x), since a vertex defines a clique of size 1; (b) edge-clique of Φ yields (1 + x) 2 as a factor of I(G • 2K 1 ; x), since an edge defines a clique of size 2; (c) clique A j ∈ Φ, |A j | ≥ 3, produces (1 + x) 2|Aj |−2 as a factor of I(G • 2K 1 ; x). Since the cliques of Φ are pairwise vertex disjoint, one can apply Corollary 3.4 to all the q cliques one by one.
w r r r r r d d d G 2 Figure 10:

Using Corollary 3.4 and the fact that
Repeating this process with {A 3 , A 4 , ..., A q }, and taking into account that all the cliques of Φ are pairwise disjoint, we obtain as required. Lemma 3.1 and Theorem 3.5 imply the following.
Corollary 3.6 [23] For every clique cover Φ of a graph G, the polynomial I(Φ(G); x) is symmetric. For an independent set S ⊂ V (G), let us denote:

Cycle covers
if an only if S ∪ W ∈ Ω G2 S , since W is an arbitrary independent set of 2qH. Hence, for each size m ≥ |S|, we get that Hence, for each positive integer m ≥ |S|, we get that Consequently, one may infer that for each size, the two graphs, G 1 and G 2 , have the same number of independent sets. In other words, I(G 1 ; x) = I(G 2 ; x).
of the antiregular graph A n is: Let us mention that I(A 2k ; x) = I(K k,k ; x) and I(A 2k−1 ; x) = I(K k,k−1 ; x), where K m,n denotes the complete bipartite graph on m+n vertices. Notice that the coefficients of the polynomial s j x j satisfy s j = s k−j for 1 ≤ j ≤ ⌊k/2⌋, while s 0 = s k , i.e., I(A 2k ; x) is "almost symmetric". Problem 4.2 Characterize graphs whose independence polynomials are almost symmetric.
It is known that the product of a polynomial P (x) = n k=0 a k x k and its reciprocal Q (x) = n k=0 a n−k x k is a symmetric polynomial. Consequently, if I(G 1 ; x) and I(G 2 ; x) are reciprocal polynomials, then the independence polynomial of G 1 ∪ G 2 is symmetric, because I (G 1 ∪ G 2 ; x) = I(G 1 ; x) · I(G 2 ; x). Problem 4.3 Describe families of graphs whose independence polynomials are reciprocal.