Partial unimodality properties of independence polynomials

Stud.Cercet.Stiint., Ser.Mat., 16 (2006), Supplement Proceedings of ICMI 45, Bacau, Sept.18-20, 2006, pp. 467-484

Partial unimodality properties of independence polynomials Vadim E. Levit and Eugen Mandrescu Abstract A stable set in a graph G is a set of pairwise non-adjacent vertices and α(G) is the size of a maximum stable set in the graph G. The polynomial I(G; x) = s0 + s1 x + s2 x2 + ... + sα xα , α = α(G), is called the independence polynomial of G (Gutman and Harary, 1983), where sk is the number of stable sets of cardinality k in G. I(G; x) is partial unimodal if the sequence of its coefficients (sk ) is partial unimodal, i.e., there are some k ≤ p such that (i) s0 ≤ s1 ≤ s2 ≤ ... ≤ sk and (ii) sp ≥ sp+1 ≥ sp+2 ≥ ... ≥ sα(G) . If k = p, then I(G; x) is called unimodal. In this paper, we survey the most important results referring the partial unimodality of independence polynomials of various families of graphs.

1

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 NG (v) = {w : w ∈ V and vw ∈ E}, and NG [v] = NG (v) ∪ {v}; if there is no ambiguity on G, we use N (v) and N [v], respectively. A vertex v is pendant if its neighborhood contains only one vertex; an edge e = uv is pendant if one of its endpoints is a pendant vertex. Kn , Pn , Cn denote respectively, the complete graph on n ≥ 1 vertices, 467

the chordless path on n ≥ 1 vertices, and the chordless cycle on n ≥ 3 vertices. By Kn1 ,n2 ,...,nq we mean the complete q-partite graph on n1 + n2 + ... + nq vertices, where ni ≥ 1, 1 ≤ i ≤ q, and if all the q parts are of the same size p, we write Kq(p) . The disjoint union of the graphs G1 , G2 is the graph G = G1 ∪ G2 having as vertex set the disjoint union of V (G1 ), V (G2 ), and as edge set the disjoint union of E(G1 ), E(G2 ). In particular, nG denotes the disjoint union of n > 1 copies of the graph G. If G1 , G2 are disjoint graphs, then their join (or Zykov sum) is the graph G1 + G2 with V (G1 ) ∪ V (G2 ) as vertex set and E(G1 ) ∪ E(G2 ) ∪ {v1 v2 : v1 ∈ V (G1 ), v2 ∈ V (G2 )} as edge set. The corona of the graphs G and H is the graph G ◦ H obtained from G and |V (G)| copies of H, such that each vertex of G is joined to all vertices of a copy of H (see Figure 1 for an example). a G

w

b

w

w

c

w

H

w

w

a

b

w w ¡ ¡ G◦H ¡ ¡ w ¡ w ¡ w

c

w @ @ w @w

Figure 1: G, H and G ◦ H. A stable set in G is a set of pairwise non-adjacent vertices. A stable set of maximum size will be referred to as a maximum stable set of G, and the stability number of G, denoted by α(G), is the cardinality of a maximum stable set in G. Let sk be the number of stable sets of cardinality k in a graph G. The polynomial I(G; x) = s0 + s1 x + s2 x2 + ... + sα xα , α = α(G), is called the independence polynomial of G, (Gutman and Harary, [13]), the independent set polynomial of G (Hoede and Li, [18]). In [11], the dependence polynomial D(G; x) of a graph G is defined as D(G; x) = I(G; −x). In [12], D(G; x) is called the clique polynomial of the graph G. For a survey on independence polynomials of graphs, see [26]. Independence polynomial was defined as a generalization of matching polynomial of a graph, [17], because the matching polynomial of a 468

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 G1 and G2 depicted in Figure 2 satisfy G2 = L(G1 ) and, hence, I(G2 ; x) = 1+6x+7x2 +x3 = M (G1 ; x), where M (G1 ; x) is the matching polynomial of the graph G1 . w

G1

c w

w

d

a e

e b¡w ©©w @

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b

w

f

w

G2

¡ ©© © w ¡ © w

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a

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@ @w

c

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Figure 2: G2 is the line-graph of and G1 . It is easy to deduce (see, for instance, [13], [2], [18]) that I(G1 ∪ G2 ; x) = I(G1 ; x) · I(G2 ; x), I(G1 + G2 ; x) = I(G1 ; x) + I(G2 ; x) − 1. The following equality, due to Gutman and Harary (see [13]) is very useful in calculating of the independence polynomial for various families of graphs. Proposition 1.1 [13] If w ∈ V (G), then I(G; x) = I(G − w; x) + x · I(G − N [w]; x). A finite sequence of real numbers (a0 , a1 , a2 , ..., an ) is said to be: • unimodal if there is some k ∈ {0, 1, ..., n}, called the mode of the sequence, such that a0 ≤ ... ≤ ak−1 ≤ ak ≥ ak+1 ≥ ... ≥ an ; • partial unimodal if there are some 0 ≤ k ≤ p ≤ n such that a0 ≤ a1 ≤ a2 ≤ ... ≤ ak and ap ≥ ap+1 ≥ ap+2 ≥ ... ≥ an ; 469

• log-concave if a2i ≥ ai−1 · ai+1 holds for i ∈ {1, 2, ..., n − 1}. It is well-known that every log-concave sequence of positive numbers is also unimodal. A polynomial is called unimodal (partial unimodal, log-concave) if the sequence of its coefficients is unimodal (partial unimodal, logconcave, respectively). Unimodal and log-concave sequences occur in many areas of mathematics, including algebra, combinatorics, and geometry (see the surveys of Brenti [5], and Stanley [37]). As a well-known example, we recall that the sequence of binomial coefficients is log-concave. For instance, I(K43 + 3K7 ; x) = 1 + 64x + 147x2 + 343x3 is unimodal, while I(K127 + 3K7 ; x) = 1 + 148x + 147x2 + 343x3 is only partial unimodal. For other examples, see [1], [23], [24], and [27]. Moreover, Alavi, Malde, Schwenk and Erd¨os proved the following theorem. Theorem 1.2 [1] For every permutation π of {1, 2, ..., α} there exists a graph G with α(G) = α such that sπ(1) < sπ(2) < ... < sπ(α) . In this paper we survey the most important findings concerning the partial unimodality of the independence polynomial of a graph.

2

Various graph families and unimodality of their independence polynomials

In the context of our paper, for instance, it is worth mentioning the following result. Theorem 2.1 [36] If ak denotes the number of matchings of size k in a graph, then the sequence of these numbers is unimodal. As a simple consequence, one can assert that the independence polynomial of any line-graph is unimodal. A graph is called claw-free if it has no induced subgraph isomorphic to K1,3 . The following result is due to Hamidoune. 470

Theorem 2.2 [16] The independence polynomial of a claw-free graph is unimodal. In [8], it was proved that for a claw-free graph G, its independence polynomial has only real roots, which ensures, according to a theorem of Newton, that I(G; x) is log-concave, and consequently, unimodal, as well.

2.1

Trees

In [2], Arocha shows that I(Pn ; x) = Fn+1 (x), where Fn (x), n ≥ 0, are the so-called Fibonacci polynomials, i.e., the polynomials defined recursively by F0 (x) = 1, F1 (x) = 1, Fn (x) = Fn−1 (x) + xFn−2 (x). Based on this recurrence, one can deduce that I(Pn ; x) =

b(n+1)/2c à n+1−j!

X

j

· xj .

j=0

As a simple application of Theorem 2.2, one can easily see that I(Pn ; x) is unimodal. In [21] and [22] we showed that I(Pn ◦K1 ; x) is unimodal, by exhibiting a claw-free graph H such that I(Pn ◦K1 ; x) = I(H; x). In [30], we proved that I(Pn ◦ 2K1 ; x) is both unimodal and palindromic (i.e., sk = sα(G)−k , 0 ≤ k ≤ bα(G)/2c). Nevertheless, for general trees, Alavi et al. stated the following (still open) conjecture. Conjecture 2.3 [1] The independence polynomial of a tree is unimodal. In support to this assertion, we found out the following result. Proposition 2.4 [28] If T is a tree with α(T ) = α, then sd(2α−1)/3e ≥ ... ≥ sα−1 ≥ sα .

471

2.2

Well-covered graphs

A graph G is said to be well-covered if every maximal stable set of G is also a maximum stable set (Plummer, [34], [35]). G is called very well-covered provided G is well-covered, without isolated vertices, and |V (G)| = 2α(G) (Favaron, [10]). For instance, the graph G = 3K10 + K120(3) is connected and wellcovered, but not very well-covered, and its independence polynomial I(G; x) = 1 + 390x + 660x2 + 1120x3 is unimodal. Brown, Dilcher and Nowakowski [6] conjectured that I(G; x) is unimodal for each well-covered graph G. Michael and Traves [33] proved that this assertion is true for every well-covered graph G having α(G) ≤ 3, while for α(G) ∈ {4, 5, 6, 7} they provided counterexamples. In [27] the following result was proved. Proposition 2.5 [27] For any integer k ≥ 4, there is a well-covered graph G with α (G) = k, whose independence polynomial is not unimodal. Nevertheless, the following conjecture is still open. Conjecture 2.6 [28] I(G; x) is unimodal for every very well-covered graph G. In [6] it was shown that every well-covered graph G on n vertices enjoys the inequalities: sk−1 ≤ k · sk and sk ≤ (n − k + 1) · sk−1 , 1 ≤ k ≤ α(G), which are strengthened as follows. Proposition 2.7 [33], [25] If G is a well-covered graph with the stability number α, then s0 ≤ s1 ≤ s2 ≤ ... ≤ sk , k = b(α + 1)/2c. In other words, for a well-covered graph G, its I(G; x) is partial unimodal with k = b(α + 1)/2c. What about the second half of the coefficients of I(G; x)? Michael and Traves proposed the following so-called ”roller-coaster ” conjecture. 472

Conjecture 2.8 [33] For each permutation π of the set {dα/2e , dα/2e+ 1, ..., α}, there exists a well-covered graph G, with α(G) = α, whose sequence (s0 , s1 , ..., sα ) satisfies sπ(dα/2e) < sπ(dα/2e+1) < ... < sπ(α) . This conjecture is still open, but the following facts are already validated. Theorem 2.9 Conjecture 2.8 is true for well-covered graphs having (i) stability numbers ≤ 7 (Michael and Traves, [33]); (ii) stability numbers ≤ 11 (Matchett, [32]). The graph G in Figure 3 is very well-covered and its independence polynomial I(G; x) = 1 + 12x + 52x2 + 110x3 + 123x4 + 70x5 + 16x6 is not only unimodal but log-concave, as well. w

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w ¡A ¡ A

w ¢ ¡ w A w ¢ ¡ ¡ A w w G @ ¢¡ XXXX A ¡ XXA¡ w @¢ w w w ¡ w @

Figure 3: A very well-covered graph with a log-concave independence polynomial. Theorem 2.10 [28] If G is a very well-covered graph of order n ≥ 2 with α(G) = α, then (i) s0 ≤ s1 ≤ ... ≤ sdα/2e and sd(2α−1)/3e ≥ ... ≥ sα−1 ≥ sα ; (ii) I(G; x) is unimodal, while α ≤ 9. In other words, we infer that for very well-covered graphs, the domain of the roller-coaster conjecture can be shortened to {dα/2e , dα/2e+ 1, ..., d(2α − 1)/3e}.

473

2.3

K¨ onig-Egerv´ ary graphs and quasi-regularizable graphs

If α(G) + µ(G) = |V (G)|, then G is called a K¨onig-Egerv´ary graph, where µ(G) is the matching number of G. According to a well-known result of K¨onig, [20], and Egerv´ary, [9], any bipartite graph is a K¨onigEgerv´ary graph. This class includes also some non-bipartite graphs (see, for instance, the graphs K3 + e and H in Figure 4). K3 + e

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w

w

wv @ H2 z w @@wy x w

Figure 4: K3 + e and H1 are K¨onig–Egerv´ary graphs, but only in H1 all µ-critical edges are also α-critical. H2 is not a K¨onig–Egerv´ary graph. Proposition 2.11 [29] If G is a K¨onig-Egerv´ ary graph with α(G) = α, then sd(2α−1)/3e ≥ ... ≥ sα−1 ≥ sα . Conjecture 2.12 [29] The independence polynomial of every K¨onig– Egerv´ary graph is unimodal. Conjecture 2.12 can not be extended to log-concavity because for every n ≥ 6 there exists a K¨onig–Egerv´ary graph, whose independence polynomial is unimodal but not log-concave. For instance, if Gn = Kn + Kn , n ≥ 6, then µ ¶ n

n

I(Gn ; x) = (1 + x)n + nx = 1 + 2nx + 2 x2 + 3 x3 + ... + xn is clearly unimodal, while s22

µ ¶2 n

− s1 s3 = 2

µ ¶ n

− 2n 3 =

´ 1 ³ 3 6n − 5n2 − n4 < 0. 12

A graph G is called quasi-regularizable if one can replace each edge of G with a non-negative integer number of parallel copies, so as to 474

obtain a regular multigraph of degree 6= 0 (see [4]). Berge proved in [4] that a graph G is quasi-regularizable if and only if |S| ≤ |N (S)| holds for every stable set S of G. The following lemma characterizes K¨onig–Egerv´ary graphs that are simultaneously quasi-regularizable. Lemma 2.13 [29] (i) A K¨onig–Egerv´ ary graph is quasi-regularizable if and only if it has a perfect matching. (ii) G is a quasi-regularizable graph of order 2α(G) if and only if it is a K¨onig–Egerv´ ary graph having a perfect matching. Let us remark that there exist quasi-regularizable graphs with nonunimodal independence polynomials, e.g., G = K10 + K6 is connected and has I(G; x) = (1 + x)6 + 10x = 1 + 16x + 15x2 + 20x3 + 15x4 + 6x5 + x6 . Corollary 2.14 [29] If G is a quasi-regularizable graph of order n = 2α(G) = 2α, then sd(2α−1)/3e ≥ ... ≥ sα−1 ≥ sα . The above inequalities are also true for very well-covered graphs, since each very well-covered graph is quasi-regularizable of order n = 2α(G) (see [4]).

2.4

Perfect graphs

A graph G is called perfect if χ(H) = ω(H) for any induced subgraph H of G, where χ(H) denotes the chromatic number of H (Berge, [3]). Proposition 2.15 [28] If G is a perfect graph with α(G) = α and ω = ω(G), then sd(ωα−1)/(ω+1)e ≥ ... ≥ sα−1 ≥ sα . The validation of the Strong Perfect Graph Conjecture, due to Chudnovsky, Robertson, Seymour and Thomas, [7], shows that C2n+1 , n ≥ 2, and C2n+1 , n ≥ 2, are the only minimal imperfect graphs. Since both 475

C2n+1 , n ≥ 2, and C2n+1 , n ≥ 2, are claw-free, we infer that the polynomials I(C2n+1 ; x), I(C2n+1 ; x) are log-concave, according to Theorem 2.2. For non-perfect graphs, Proposition 2.15 is not necessarily false; for example, I(C7 ; x) = 1 + 7x + 14x2 + 7x3 . However, there are imperfect graphs, whose independence polynomials are not unimodal. For example, if H = K97 + 4K3 , and G is the graph obtained from H by adding an edge that joins a vertex of K97 to a vertex of some C5 , then G is a connected imperfect graph whose independence polynomial is not unimodal, because I(G; x) = 1 + 114x + 603x2 + 921x3 + 891x4 + 945x5 + 405x6 . Since each bipartite graph G is perfect and has ω(G) ≤ 2, we obtain the following result. Corollary 2.16 [28] If G is a bipartite graph with α(G) = α ≥ 1, then sd(2α−1)/3e ≥ ... ≥ sα−1 ≥ sα . In particular, a similar result is true for trees, whose importance is significant vis-`a-vis the conjecture of Alavi et al.

3

Compound Graphs

It is easy to see that G ◦ K1 is a very well-covered graph, for any graph G. In [22] we showed that I(Pn ◦ K1 ; x) is unimodal, while for the general case it is known the following result. Theorem 3.1 (i) [24] If G is a graph of order n and α(G) ≤ 3, then I(G ◦ K1 ; x) is log-concave with ¹

º

¹

º

n+1 n+1 ≤ mode(G ◦ K1 ) ≤ + 1. 2 2

In particular, if α(G) = 2 and n is odd, or α(G) = 1, then ¹

º

n+1 . mode(G ◦ K1 ) = 2 476

(ii) [23] If G is a graph of order n and α(G) = 4, then I(G ◦ K1 ; x) is unimodal with ¹

º

¹

º

¹

º

n+1 n+1 ≤ mode(G ◦ K1 ) ≤ + 2. 2 2

Moreover, if n is odd, then ¹

º

n+1 n+1 ≤ mode(G ◦ K1 ) ≤ + 1. 2 2

A polynomial

n P i=o

ci xi is called palindromic if ci = cn−i , i = 0, 1, ..., bn/2c.

The palindromicity of matching polynomial and characteristic polynomial of a graph are examined in Kennedy [19], while for independence polynomial we quote Gutman [14], Gutman [15], and Stevanovi´c [38]. Recall here, from Stevanovi´c [38], the following two ways to construct graphs having palindromic independence polynomials. • Rule 1: ”Pendant vertex construction”. For a given graph H, define a new graph G as: G = H ◦ 2K1 . For instance, the graphs in Figure 5 have: I(H; x) = 1 + 6x + 9x2 + 2x3 , and I(G; x) = 1+15x+90x2 +290x3 +565x4 +702x5 +565x6 +290x7 +90x8 +15x9 +x10 .

w

H

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@ @w

w

G

w

w w w ¡@ ¡ ¡ ¡ @ w w @¡ w w ¡ ¡ ¡ ¡ ¡ w w w ¡ w ¡

w w w

Figure 5: H and G = H ◦ 2K1 . A cycle cover of a graph H is a spanning graph of H, each connected component of which is a vertex (called a vertex-cycle), an edge (called an edge-cycle), or a proper cycle. 477

Proposition 3.2 [31] If G = H ◦ 2K1 has α(G) = α, then I(G; x) is palindromic and its coefficients sk satisfy s0 ≤ s1 ≤ ... ≤ sp , p = b(2α + 2)/5c, and st ≥ st+1 ≥ ... ≥ sα−1 ≥ sα , t = b(3α − 2)/5c. • Rule 2: ”Cycle cover construction”. Let Γ be a cycle cover of H. Construct a new graph, denoted by G = Γ{H}, as follows. If C ∈ Γ is: (i) a vertex-cycle, say v, then add two non-adjacent vertices and join them to v; (ii) an edge-cycle, say uv, then add two non-adjacent vertices and join them to both u and v; (iii) a proper cycle, with V (C) = {vi : 1 ≤ i ≤ s}, E(C) = {vi vi+1 : 1 ≤ i ≤ s−1}∪{v1 vs }, then add s pairwisw non-adjacent vertices, say {wi : 1 ≤ i ≤ s} and each of them is joined to two consecutive vertices on C, as follows: w1 is joined to vs , v1 , then w2 is joined to v1 , v2 , further w2 is joined to v2 , v3 , etc.

H

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w

w

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w w ¡@ ¡ @ ¡ w w @w ¡ ¡ w ¡

w @

w

w

w

@ @w ¡ ¡

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w

Figure 6: H and some G, built according Rule 2. For an example, see the graphs in Figure 6: I(H; x) = 1 + 6x + 9x2 + 2x3 , and I(G; x) = 1 + 13x + 60x2 + 124x3 + 124x4 + 60x5 + 13x6 + x7 . Proposition 3.3 [31] If H is a graph of order n ≥ 2, Γ is a cycle cover of H that contains no vertex-cycles, G = Γ{H}, α(G) = α, then the coefficients sk of I(G; x) satisfy: s0 ≤ s1 ≤ ... ≤ sp , p = b(α + 1)/3c, and sq ≥ sq+1 ≥ ... ≥ sα−1 ≥ sα , q = b(2α − 1)/3c. 478

A clique cover of a graph G is a spanning graph of G, each component of which is a clique. • Rule 3: ”Clique cover construction”. Let Ω be a clique cover of H. construct a new graph G from H, denoted by G = Ω{H}, as follows: for each clique Q ∈ Ω, add two new nonadjacent vertices and join them to all the vertices of Q. Figure 7 contains an example: Ω = {{a, b, c}, {d, e}, {f }} is a clique cover of G that has a clique consisting of one vertex. Ga

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@ c @w

dw

e

w

w w w H @HH@ ¡ H ¡ @ H@ @w @w ¡ HH

f

w

w w @ ¡ ¡ @ w ¡ @w

w w

w

Figure 7: G and H = Ω{G}. The independence polynomials of G and H = Ω{G}, from Figure 7, are I(G; x) = 1 + 6x + 9x2 + 2x3 , I(H; x) = 1 + 12x + 48x2 + 76x3 + 48x4 + 12x5 + x6 , but only I(H; x) is palindromic. Let Hn , n ≥ 1, be the graphs obtained according to Rule 3 from Pn , as one can see in Figure 8. w

v H2n−1 w w

P2n−1 w

w w ¡ ¡ ¡ ¡ ¡ rrrrrw ¡ w w w @ @ @ @ @w @w

w ¡ v¡¡ w H2n w @ @ @w

w

P2n

w rrrrrw

w

w

w

w w ¡ ¡ ¡ ¡ ¡ rrrrrw ¡ w w w @ @ @ @ @w @w w

w rrrrrw

w

Figure 8: Pn and Hn = Ω{Pn }. Theorem 3.4 [30] (i) I(Hn ; x) is both palindromic and unimodal. (ii) I(Pn ◦ 2K1 ; x) is palindromic and unimodal. Moreover, if a clique cover Ω of Pn contains m vertices as cliques, then I(Pn ◦2K1 ; x) = (1 + x)n−m I(Ω{Pn }; x). 479

4

Conclusions

We have summarized a number of important findings connecting the partial unimodality phenomena with independence polynomials of graphs. We also compiled a list of conjectures offering opportunities for synthesis of both combinatorial and algebraic methods.

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[31] V. E. Levit, E. Mandrescu, Partial unimodality for independence polynomials of some compaund graphs, Journal of the Faculty of Electrical Engineering-series Mathematics, Belgrade (2006) (submitted). [32] P. Matchett, Operations on well-covered graphs and the RollerCoaster Conjecture, Electronic Journal of Combinatorics, 11 #45 (2004). [33] T. S. Michael, W. N. Traves, Independence sequences of wellcovered graphs: non-unimodality and the Roller-Coaster conjecture, Graphs and Combinatorics 19 (2003) 403-411. [34] M. D. Plummer, Some covering concepts in graphs, Journal of Combinatorial Theory 8 (1970) 91-98. [35] M. D. Plummer, Well-covered graphs – a survey, Questiones Mathematicae 16 (1993) 253-287. [36] A. J. Schwenk, On unimodal sequences of graphical invariants, Journal of Combinatorial Theory B 30 (1981) 247-250. [37] R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Annals of the New York Academy of Sciences 576 (1989) 500-535. [38] D. Stevanovic, Graphs with palindromic independence polynomial, Graph Theory Notes of New York Academy of Sciences XXXIV (1998) 31-36. Department of Computer Science Holon Institute of Technology 52 Golomb Str., P.O. Box 305 Holon 58102, ISRAEL and Department of Computer Science and Mathematics The College of Judea and Samaria Ariel 44837, ISRAEL [email protected] 483

Department of Computer Science Holon Institute of Technology 52 Golomb Str., P.O. Box 305 Holon 58102, ISRAEL eugen [email protected]

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