Discrete Applied Mathematics the Independence Polynomial of Rooted

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Discrete Applied Mathematics 158 (2010) 551–558

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Discrete Applied Mathematics

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The independence polynomial of rooted products of graphs Vladimir R. Rosenfeld ∗ Institute of Evolution, University of Haifa, Mount Carmel, Haifa 31905, Israel article info a b s t r a c t

Article history: A stable (or independent) set in a graph is a set of pairwise nonadjacent vertices thereof. Received 25 October 2007 The stability number α(G) is the maximum size of stable sets in a graph G. The independence Received in revised form 13 July 2009 polynomial of G is Accepted 17 October 2009 Available online 31 October 2009 α(G) X k 2 α(G) I(G; x) = skx = s0 + s1x + s2x + · · · + sα(G)x (s0 := 1), Keywords: k=0 Independence polynomial where s is the number of stable sets of cardinality k in a graph G, and was defined by Rooted product of graphs k Real independence-polynomial roots Gutman and Harary (1983) [13]. We obtain a number of formulae expressing the independence polynomials of two sorts of the rooted product of graphs in terms of such polynomials of constituent graphs. In particular, it enables us to build infinite families of graphs whose independence polynomials have only real roots. © 2009 Elsevier B.V. All rights reserved.

1. Introduction

Graph invariants and graph polynomials are widely studied. The independence polynomial has a long history, dating back to a paper by Harary and Gutman [13] in 1983. The characteristic polynomial and the matching polynomial found applications in chemistry and physics. The independence polynomial was introduced as an analog of the matching polynomial [13]. Classical questions about these graph polynomials concern their computation and the existence of real roots. For computations, the compositional behavior of these polynomials is intensively studied. The simplest case of compositionality is multiplicativity, which states that the polynomial of a disjoint union of two graphs equals the product of the polynomial computed for each graph separately. It is of interest whether other graph operations (products, join, etc.) can be computed similarly. Godsil and MacKay [12] introduced the rooted product and showed its compositional behavior for the characteristic polynomial. The method of computing such polynomials using a transfer matrix for certain regularly constructed graphs is based on certain compositionalities. So is the computation of the matching and independence polynomial for graphs of bounded tree-width. A general discussion of such compositionality theorems may be found in Makowsky [27]. There, the existence of compositionality theorems is proven for a wide class of graph operations. However, finding explicit and fairly simple compositionality theorems remains often a challenge. The rooted products, it should be noted, are not covered by the methods described in the above-mentioned paper. The results obtained in the present paper show how this can be done using different manipulations with polynomials. The question of real roots has interpretations in physical applications. Herein, we also deal with such roots. The matching polynomial of a graph has only real roots. But the matching polynomial of a graph G equals the independence polynomial of the line graph L(G) of G. So, this is also true for the independence polynomial for line graphs. Line graphs are claw-free

∗ Corresponding address: Mathematical Chemistry Group, Department of Marine Sciences, Texas A&M University at Galveston, Galveston, TX 77553–1675, USA. Tel.: +1 (409) 740 4538; fax: +1 (409) 457 0426. E-mail addresses: [email protected], [email protected], [email protected].

Fig. 1. G2 is the line graph of G1. graphs, and it was asked by Hoede and Li [17] and conjectured by Stanley [34] that this holds in general for claw-free graphs. The conjecture was recently confirmed by Chudnovsky and Seymour in [6]. The present paper discusses how to compute the independence polynomial I(G; x), also called stable set polynomial in the literature, for two versions of rooted products G ◦ H and G E◦ H of two graphs G = (V (G); E(G)) and H = (V (H); E(H), v), where v is the root of H. The main results are two compositionality theorems, Theorems 7 and 10. Both express I(G ◦ H; x), respectively I(G E◦ H; x), as an algebraic expression in I(H − v; x), I(H − N[v]; x), and a substitution instance I(G, f1(I(H − v; x), I(H − N[v]; x))), respectively I(G, f2(I(H − v; x), I(HN [v]; x))). As an application of these formulae, the question for which graphs G the polynomial I(G; x) has only real roots is discussed. Propositions 8 and 11 show that if I(G; x) has only real roots, this is the case also for the independence polynomials I(G ◦ Kn; x) and I(G E◦ Kn; x) of the rooted products G ◦ Kn and G E◦ Kn, respectively, which will be defined below. This gives a method to produce infinitely many examples of graphs which have an induced claw (a K1,3) and for which still all roots of the independence polynomial are real.

2. Preliminaries

Throughout the paper, we consider just finite, undirected graphs without multiple edges and self-loops. Let G = (V , E) be a graph with the vertex set V = V (G) and edge set E = E(G). If X ⊆ V , then G[X] is the induced subgraph, of G, spanning 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 write 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 in G, we use N(v) and N[v], respectively. Moreover, Kn, Pn, and Cn denote consecutively the complete graph on n ≥ 1 vertices, the chordless path on n ≥ 1 vertices, and the chordless cycle on n ≥ 3 vertices. A stable (or independent) set in G is a set of pairwise nonadjacent vertices of G. A stable set of the 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 maximum cardinality of stable sets in G. Let sk be the number of stable sets of cardinality k in a graph G. The polynomial α X k 2 α I(G; x) = skx = s0 + s1x + s2x + · · · + xαx (s0 := 1; α = α(G)) (1) k=0 is called the independence polynomial of G [13], or the independent set polynomial of G [17]; see also [1–6,14,15,19–27,32–34]. The independence polynomial was defined as a generalization of the matching polynomial of a graph [13], because the matching generating polynomial of a graph G and the independence polynomial of its line graph L(G) 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 iff (if and only if) they share an end point in G. For instance, the graphs G1 and G2 depicted in Fig. 1 satisfy G2 = L(G1) and, hence, 2 3 I(G2; x) = 1 + 6x + 7x + x = M(G1; x), where M(G1; x) is the matching generating polynomial of the graph G1. Notice that the independence polynomial is also applied to solve different problems of (mathematical) chemistry [16], statistical physics [32,33], and, potentially, of many other fields. The following equality, due to Gutman and Harary [13], is very useful for evaluating the independence polynomial for various families of graphs.

Proposition 1 (Gutman and Harary). If w ∈ V (G), then I(G; x) = I(G − w; x) + xI(G − N[w]; x). (2)

In what follows, we also use the inverse independence polynomial, defined by the equality: 1 ϕ(G; x) = xnI G; (n = |V (G)|). (3) x Clearly, the inverse relation is 1 I(G; x) = xnϕ G; (n = |V (G)|). (4) x

If V (G) = {ui | 1 ≤ i ≤ n} and to each vertex ui we associate a variable xi(1 ≤ i ≤ n) (see [11,28–31]), then both I(G; x) and ϕ(G; x) can also be represented in the respective n-variable forms, namely, I(G; x) and ϕ(G; x), where V.R. Rosenfeld / Discrete Applied Mathematics 158 (2010) 551–558 553

Fig. 2. The graph G has α(G) = 2.

x = (x1, x2,..., xn). (We consider below the number of components in x as a dynamic parameter that depends on the graph.) For example, the graph from Fig. 2 has I(G; x) = 1 + 4x + 2x2,

I(G; x) = 1 + (x1 + x2 + x3 + x4) + (x1x3 + x1x4), 1 ϕ(G; x) = x4I G; = x4 + 4x3 + 2x2, x

ϕ(G; x) = x1x2x3x4 + (x2x3x4 + x1x3x4 + x1x2x4 + x1x2x3) + (x2x3 + x2x4). Lemma 2 shows the utility of the notation ϕ(G; x) (see also [11,28–33]).

= { } ⊆ Lemma 2. If A ui1 , ui2 ,..., uiq V (G), then ∂ q ϕ(G − A; x) = ϕ(G; x). (5) ··· ∂xi1 ∂xi2 ∂xiq

∂ Proof. The equality follows from the fact that ϕ(G − ui; x) = ϕ(G; x) holds for each ui ∈ V (G). ∂xi

∂2 As a case in point, ϕ(G − {u1, u3}; x) = ϕ(G; x) = x2x4 + x2 + x4, where G is depicted in Fig. 2. ∂x1∂x3 For a graph G = (V , E) of order n, let us denote X ψq(G; x) = ϕ(G − A; x)(0 ≤ q ≤ n) and (6) A⊆V ;|A|=q Xn X Ψ (G; x) = ψq(G; x) = ϕ(G − A; x). (7) q=0 ∅⊆A⊆V Consequently, we have also ; = ; | ψq(G x) ψq(G x) x1=x2=···=xn=x and (8) ; = ; | Ψ (G x) Ψ (G x) x1=x2=···=xn=x. (9)

Lemma 3 ([11]). If G = (V , E) is a graph of order n, then (i) for every q ∈ {0, 1,..., n} we have 1 dq ψq(G; x) = ϕ(G; x); (10) q! dxq (ii) the total sum can be expressed as ∞ Xn 1 dq X 1 dq Ψ (G; x) = ϕ(G; x) = ϕ(G; x) q! dxq q! dxq q=0 q=0 d = exp ϕ(G; x) = ϕ(G; x + 1). (11) dx

Let G and H be two graphs of order n1 and n2, respectively, and u ∈ V (G), v ∈ V (H). By G H we mean the graph having V (G) ∪ V (H) as the vertex set and E(G) ∪ E(H) ∪ uv as the edge set, while its order n = n1 + n2. Now, using Proposition 1 (borrowed from [13]), we obtain I(G H; x) = I(G − u; x) · I(H − v; x) + x · I(G − u; x) · I(H − N[v]; x). (12) 554 V.R. Rosenfeld / Discrete Applied Mathematics 158 (2010) 551–558

Fig. 3. G = P3 E◦ K1,2, where V (P3) = {u1, u2, u3} and v is the root of K1,3.

Hence, we get: 1 1 − 1 ϕ(G H; x) = xn · I G H; = x · xn1 · I G; · xn2 1 · I H − v; x x x 1 − 1 −| | 1 + · xn1 1 · I G − u; · xn2 N(v) · I H − N[v]; x x x | [ ]| = x · ϕ(G; x) · ϕ(H − v; x) + x N v · ϕ(G − u; x) · ϕ(H − N[v]; x) | | = x · ϕ(G; x) · ϕ(H − v; x) + x NG H (v) · ϕ(G − u; x) · ϕ(H − N[v]; x), (13) which can be written using n variables as follows (see [11,28–31]): ! Y ϕ(G H; x) = xv · ϕ(G; x) · ϕ(H − v; x) + xw · ϕ(G − u; x) · ϕ(H − N[v]; x), (14)

∀w∈NG H (v) where the product Q embraces all variables pertaining to vertices that are adjacent to v in G H. ∀w∈NG H (v) Now, using Lemma 2, we can write (by analogy with [11,30,31]): ( ! ) Y ∂ ϕ(G H; x) = xvϕ(H − v; x) + xw ϕ(H − N[v]; x) ϕ(G; x) ∂xu ∀w∈NG H (v)   !   Q   xw ϕ(H − N[v]; x)    ∀ ∈    w NG H (v) ∂  = [x ϕ(H − v; x)] 1 + ϕ(G; x) v  − ;    xvϕ(H v x) ∂xu       ∂ = αu(x) 1 + βu(x) ϕ(G; x), (15) ∂xu where

αu(x) = xvϕ(H − v; x) and ! Q xw ϕ(H − N[v]; x) ∀w∈NG H (v) βu(x) = . xvϕ(H − v; x) It is very important to additionally emphasize that the unequal indices u (of a vertex in G) and v (of a vertex in H) necessarily belong to the edge that joins G to H in G H, and, besides, u is the only vertex of G that belongs to NG H (v) (among vertices w therein). Let V (G) = {u1 | 1 ≤ i ≤ n} and v ∈ V (H). The rooted product G ◦ H of a graph G and a rooted graph H with respect to the ‘‘root’’ v is defined as follows (see p. 118 in [7] and [12,17,18]): take |V (G)| copies of H, and for every vertex vi of G, identify vi with the root v of the ith copy of H; |V (G ◦ H)| = |V (G)||V (H)| and |E(G ◦ H)| = |E(G)| + |V (G)||E(H)|. A version of this is another product G E◦ H where every vertex ui of G is joined with an edge uivi to the root vi of respective copy Hi; |V (GE◦H)| = |V (G)| + |V (G)||V (H)| and |E(G E◦ H)| = |E(G)| + |V (G)| + |V (G)||E(H)| (see Fig. 3 for an example). Here, we turn to the main part of our report. We obtain a number of formulae that express the independence polynomial I(G E◦ H; x) of the rooted product G E◦ H, of graphs G and H, in terms of the polynomials I(G; x) and I(H; x) of the initial graphs. The parallel results for I(G ◦ H; x) are given without proofs, since deriving them is done in a similar way. This, in particular, enables us to prove that, for the complete graph Kp on p ≥ 1 vertices and every graph G whose independence polynomial I(G; x) has only real roots, the polynomial I(G E◦ Kp; x) has only real roots. Due to the last, included results, the conclusion follows that the same holds true for I(G ◦ H; x) as well. In this work, we utilize our previous approaches based on using differential operators in many (vertex) variables [11,28–31]. V.R. Rosenfeld / Discrete Applied Mathematics 158 (2010) 551–558 555

3. The main part

We begin this section with quite a technical lemma which plays, however, a crucial part below.

Lemma 4. Let f (x1, x2,..., xn) be an arbitrary function linear to every variable xi (1 ≤ i ≤ n) and let γ be a constant. Then ∂ ∂ 1 + γ f (x1, x2,..., xn) = exp γ f (x1, x2,..., xn)(1 ≤ i ≤ n). (16) ∂xi ∂xi

Now, using this fact, (15) and the definition of the rooted product G E◦ H above, we can state the following lemma dealing with its inverse independence polynomial ϕ(G E◦ H; x).

Lemma 5. Let ϕ(G E◦ H; x) be the inverse independence polynomial of the rooted product G E◦ H(|V (G E◦ H)| = |V (G)| + |V (G)||V (H)|; |E(G E◦ H)| = |E(G)| + |V (G)| + |V (G)||E(H)|; |V (G)| = n1; |V (H)| = n2). Then

" n #( n " #) Y1 Y1 ∂ ϕ(G E◦ H; x) = αu (x) 1 + βu (x) ϕ(G; x) j j x j=1 j=1 ∂ uj " n #( n ) Y1 Y1 ∂ = αu (x) exp βu (x) ϕ(G; x) j j x j=1 j=1 ∂ ui " n # " n # Y1 X1 ∂ = αu (x) exp βu (x) ϕ(G; x) j j x j=1 j=1 ∂ uj

" n1 # = Y ; + ; + ; ; + αuj (x) ϕ(G x1 β1(x) x2 β2(x) ... xn1 βn1 (x)). (17) j=1

Proof. First, note that the construction of G E◦ H uses the n1-tuple application of the binary operation on graphs, involving here a core graph G and n1 copies of a rooted graph H. For this reason, the second part of (16) looks like the product of n1 third parts of (15), where the factors differ just in the indices of variables pertaining to respective vertices of constituent graphs.

Further, since all derivatives in (16) are just with respect to G-variables, every term αuj (x) (which involves just Hj-variables) behaves as a constant in the formula. Besides, as was mentioned above, each term βuj (x) contains (among NG E◦ H -variables) ∂ just only one xu ; thus, βu (x) = 0 ∀i 6= j. Hence, every term βu (x) also behaves as a constant in (16), serving as a j ∂xi j j ∂ coefficient before x . All this proves the first equality in (16). The second equality is due to the technical Lemma 4 above. ∂ uj The third equality is due to the elementary fact that ea · eb = ea+b for arbitrary quantities a and b. Lastly, the remaining, fourth, equality also follows from familiar facts in calculus. This completes the proof. Note that the analytical forms of the second to fifth parts of (17) are also true in the more general cases of the rooted = { } products above where the family Γ H1, H2,..., Hn1 of not necessarily isomorphic graphs is used instead of n1 isomorphic copies of one rooted graph H (see p. 118 in [7] and [11,30,31]). Much the same arguments using individual vertex variables and respective differential operators were previously employed by us in [11,28–31] but, herein, some proofs are done in a more direct way. Since using many variables plays only a subsidiary role in this paper, we return below to the one-variable mathematics. To this end, we first note a corollary of Lemma 5, viz.:

E◦ ; = E◦ ; | Corollary 5.1. Let ϕ(G H x) ϕ(G H x) x1=x2=···=xn=x be a one-variable, inverse independence polynomial of the rooted product G E◦ H of graphs G and H. Then

ϕ(G E◦ H; x) = [α(x)]n1 ϕ[G; x + β(x)], (18) = | = | where α(x) αuj (x) x1=x2=···=xn=x and β(x) βuj (x) x1=x2=···=xn=x. Note in passing that the last side of (15), (17) and (18) may play an important role also in calculating other graph polynomials (see [11,28–31]), whenever one can give the respective interpretation to expressions α(x) and β(x). However, it is beyond the scope of the present work. By virtue of Corollary 5.1, we can derive Lemma 6.

Lemma 6. Let ϕ(G E◦ H; x) be the inverse independence polynomial of the rooted product G E◦ H of graphs G and H. Then 556 V.R. Rosenfeld / Discrete Applied Mathematics 158 (2010) 551–558

| [ ]| x NH v ϕ(H − N[v]; x) ϕ(G E◦ H; x) = [xϕ(H − v; x)]n1 ϕ G; x + xϕ(H − v; x) xϕ(H − v; x) + xd−1ϕ(H − N[v]; x) = [xϕ(H − v; x)]n1 ϕ G; , (19) ϕ(H − v; x) where d is the degree (valency) of a vertex v in the rooted product G E◦ H or, in other terms, d = |NG E◦ H (v)| = |NH [v]| = |NH (v)| + 1. The following corollary will play a crucial role in studying the roots of the independence polynomial I(G E◦ H; x) below (as well as directly of ϕ(G E◦ H; x)).

Corollary 6.1. Let ϕ(G E◦ H; x) be the inverse independence polynomial of the rooted product G E◦ H of graphs G and H (as above). ; Besides, let λ1, λ2, . . . , λn1 be the roots of the inverse independence polynomial ϕ(G x) of G. Then

n1 xϕ(H − v; x) + x|N(v)|ϕ(H − N[v]; x) n1 Y ϕ(G E◦ H; x) = [xϕ(H − v; x)] − λj H − ; x j=1 ϕ( v )

n1 Y 2 |N[v]| = x ϕ(H − v; x) + x ϕ(H − N[v]; x) − λjxϕ(H − v; x) . (20) j=1

; = Qn1 − Proof. The first equality follows from the formula I(G x) j=1(x λj), after replacing x in parentheses with the respective expression. The second equality is elementary, which completes the proof. Lemma 6 allows us to state here the main analytical result for the independence polynomial I(G E◦ H; x), viz.:

Theorem 7. Let I(G E◦ H; x) be the independence polynomial of the rooted product G E◦ H of graphs G and H. Then xI(H − v; x) I(G E◦ H; x) = [I(H − v; x) + x · I(H − N[v]; x)]n1 · I G; I(H − v; x) + xI(H − N[v]; x)

xI(H − v; x) n1 n1 Y = [I(H; x)] · I G; = I(H; x) − λjxI(H − v; x) , (21) I H; x ( ) j=1 where λj is the jth root of the inverse independence polynomial ϕ(G; x) of a graph G. E◦ ; = n E◦ ; 1 = | E◦ | Proof. The first equality, in (21), is due to the fact that I(G H x) x ϕ(G H x )(n V (G H) ), while the second one is by virtue of Proposition 1. The last, third, equality is proven by analogy with Corollary 6.1, which completes the proof. Theorem 7 has Corollary 7.1:

− Corollary 7.1. The polynomial [I(H; x)]n1 α(G) is a divisor of the polynomial I(G E◦ H; x).

Proof. By definition (see (3)), ϕ(G; x) has n1 − α(G) zero eigenvalues. Taking into account the last side of (21), where eigenvalues of ϕ(G; x) are involved, we arrive at the proof.

For example, for G = K2 and H = K1 we have x I(G E◦ H; x) = (1 + x)2I G; 1 + x 2x = (1 + x)2 1 + 1 + x = (1 + 3x)(1 + x) = 1 + 4x + 3x2. Our next proposition practically enables one to construct families of graphs whose independence polynomials have only real roots.

Proposition 8. Let G be a graph whose independence polynomial I(G; x) has only real (namely, negative) roots, and let Kp be the complete graph on p ≥ 1 vertices. Then the polynomial I(G E◦ Kp; x) has only real roots, which are all negative. Proof. It is enough to prove a similar statement for ϕ(G E◦ H; x), noting that ϕ(G; x) has only real roots, which follows from p−1 p−2 so properties of I(G; x). Substitute the polynomial ϕ(Kp−1; x) = x + (p − 1)x for ϕ(H − v; x) and 1 for ϕ(H − N[v]; x) on the third side of (20); we obtain V.R. Rosenfeld / Discrete Applied Mathematics 158 (2010) 551–558 557

n1 Y 2 p−1 p−2 p p−1 p−2 ϕ(G E◦ Kp; x) = x x + (p − 1)x + x · 1 − λjx x + (p − 1)x j=1

n1 n1(p−1) Y 2 = x x + (p − λj)x − λj(p − 1) . (22) j=1

In other words, we have a system of n1 quadratic equations with real coefficients. Since the general equation form is 2 x + (p − λj)x − λj(p − 1) = 0, its solutions are

p 2 p 2 −(p − λj) ± (p − λj) + 4λj(p − 1) −(p − λj) ± (p + λj) − 4λj x1,2 = = . 2 2 √ 2 Evidently, for all real values of λj, the expression under is > 0. Indeed, for all p ≥ 1, it is easy to see that (p − λj) + 2 4λj(p − 1) ≥ 0 if λj ≥ 0; and, similarly, (p + λj) − 4λj ≥ 0 if λj ≤ 0. Since (p − λj) is also a real number, so is x1,2, as well. By definition (1), the independence polynomial of any graph does not have negative coefficients. According to Descartes’ rule of signs, if such a polynomial has real roots, all these roots are negative. Whence we immediately arrive at the overall proof. = { E◦ }∞ For every graph G obeying the conditions of Proposition 8, there exist the infinite series Sp (G Kp)s s=1 of iterated rooted products, where p ≥ 1; (G E◦ Kp)1 := G E◦ Kp and (G E◦ Kp)s+1 = [(G E◦ Kp)s E◦ Kp] (s ≥ 1). Similar graphs were termed F-graphs by Farrell [10] (see also [8,9]); their different spectral properties were previously studied in [30] (and the roots in [2–4,6,20,21,23,25,26,31]). Besides these, we know some other infinite families of graphs with such spectral properties. As close examples, one may choose ‘irregular’ series S where not necessarily equal complete graphs Kp (with dynamic p) are used at different consecutive stages of iterating members of the series. Apparently, there exist infinitely many series of this sort. Moreover, very similar families can be built, by analogy, using the classical rooted product G ◦ H (see p. 118 in [7] and [11,30,31]), as follows from the results which are additionally given below. We do not here present our proofs, since these were done in the same universal way as we already considered above.

Lemma 9. Let ϕ(G ◦ H; x) be the inverse independence polynomial of the rooted product G ◦ H of graphs G and H. Then | | n1 xϕ(H − v; x) ϕ(G ◦ H; x) = x N(v) ϕ(H − N[v]; x) ϕ G; . (23) x|N(v)|ϕ(H − N[v]; x)

Corollary 9.1. Let ϕ(G ◦ H; x) be the inverse independence polynomial of the rooted product G ◦ H of graphs G and H. Then

n1 Y |N(v)| ϕ(G ◦ H; x) = xϕ(H − v; x) − λjx ϕ(H − N[v]; x) . (24) j=1

Theorem 10. Let I(G ◦ H; x) be the independence polynomial of the rooted product G ◦ H of graphs G and H. Then

xI(H − N[v]; x) n1 n1 Y I(G ◦ H; x) = [I(H − v; x)] I G; = [I(H − v; x) − λjxI(H − N[v]; x)], (25) I H − ; x ( v ) j=1 where λj is the jth root of the inverse independence polynomial ϕ(G; x) of a graph G.

− Corollary 10.1. The polynomial [I(H − v; x)]n1 α(G) is a divisor of the polynomial I(G ◦ H; x).

Proposition 11. Let G be a graph whose independence polynomial I(G; x) has only real (namely, negative) roots, and let Kp be the complete graph on p ≥ 1 vertices. Then the polynomial I(G ◦ Kp; x) has only real roots, which are all negative.

4. Conclusions

From the obtained results, it follows that sets of both rooted products G◦H and G E◦ H, of graphs G and H, contain infinitely many examples of graphs which have an induced claw (a K1,3) and for which still all roots of the independence polynomial are real (namely, negative). Thus, line graphs and claw-free graphs are not the only graphs having such roots. Corollaries 7.1 and 10.1 establish that a necessary condition for polynomials I(G ◦ H; x) and I(G E◦ H; x) to have all roots real is that the polynomials I(H − v; x) and I(H; x) of graphs H − v and H, respectively, must also have only real roots. Similar conditions should be imposed on the polynomial I(G; x) of a core graph G. In general, one has to resolve n1 equations of the form I(H−v; x)−λjxI(H−N[v]; x) = 0 (res. I(H; x)−λjxI(H−v; x) = 0) which all together produce the roots of the independence polynomial I(G◦H; x) (res. I(GE◦H; x)), in order to establish if these are all real. Such a task can be performed using Maple or similar program packages. 558 V.R. Rosenfeld / Discrete Applied Mathematics 158 (2010) 551–558

The mentioned topic raises, in particular, the following question: Do there exist analytical methods allowing one to predict common roots, if any, for polynomials I(H1; x) and I(H2; x)? Herein, it is especially interesting in cases of H1 = H, H2 = H − v and H1 = H − v, H2 − N[v], which are considered in Theorems 7 and 10, respectively. Progress in solving this question may help to find other series of graphs possessing real independence-polynomial spectra. Lastly, we would like to emphasize the role of using many (vertex) variables, which are undoubtedly advantageous in the intermediate analytical manipulations displayed by the expressions (5)–(7) and (14)–(17). See also [11,28–31] and two recent papers by Scott and Sokal [32,33], who discussed also the benefits from using many variables instead of just one variable x.

Acknowledgements

I thank Profs. Eugen Mandrescu (Holon, Israel), Francisco Torrens (Valencia, Spain), and Douglas J. Klein (Galveston, USA) for their generous help and attention to the present work. Recommendations of the referees enabled the author to considerably improve the presentation of material and are highly appreciated.

References

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