Distinctive power of the alliance polynomial forregular graphs

Walter Carballosa 1

Facultad de Matematicas, Universidad Autonoma de Guerrero

Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico

Jose M. Rodrıguez 2

Departamento de Matematicas, Universidad Carlos III de Madrid

Av. de la Universidad 30, 28911 Leganes, Madrid, Spain

Jose M. Sigarreta 3

Facultad de Matematicas, Universidad Autonoma de Guerrero

Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico

Yadira Torres-Nunez 4

Departamento de Matematicas, Universidad Carlos III de Madrid

Av. de la Universidad 30, 28911 Leganes, Madrid, Spain

Abstract

The alliance polynomial of a graph G with order n and maximum degree ∆ is thepolynomial A(G;x) =

∑∆k=−∆Ak(G)xn+k, where Ak(G) is the number of exact

defensive k-alliances in G. We obtain some properties of A(G;x) and its coefficientsfor regular graphs. In particular, we characterize the degree of regular graphs bythe number of non-zero coefficients of their alliance polynomial. Besides, we provethat the family of alliance polynomials of ∆-regular graphs with small degree is a

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Electronic Notes in Discrete Mathematics 46 (2014) 313–320

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very special one, since it does not contain alliance polynomials of graphs which arenot ∆-regular.

Keywords: Finite Graphs, Regular Graphs, Defensive Alliances, AlliancePolynomials.

1 Introduction

Some parameters of a graph allow to define polynomials on the graph, forinstance, the parameters associated to matching sets [8,9], independent sets[4,10,11], domination sets [1,3,2], chromatic numbers [12,14] and many others.In [6], the authors use the exact index of alliance (introduced in [5]) in orderto define the exact alliance polynomial of a graph. The main aim of this workis to obtain further results about the alliance polynomial of regular graphs,since they are a very interesting class of graphs.

We begin by stating the terminology. Throughout this paper, G = (V,E)denotes a simple graph (no necessarily connected) of order |V | = n and size|E| = m. We denote two adjacent vertices u and v by u ∼ v. For a nonemptyset X ⊆ V and a vertex v ∈ V , NX(v) denotes the set of neighbors v has inX: NX(v) := {u ∈ X : u ∼ v}, and the degree of v in X will be denoted byδX(v) = |NX(v)|. We denote by δ and ∆ the minimum and maximum degreeof G, respectively. If G is a regular graph with degree ∆ we say that G is∆-regular. The subgraph induced by S ⊂ V will be denoted by 〈S〉 and thecomplement of the set S ∈ V will be denoted by S.

A nonempty set S ⊆ V with 〈S〉 connected is a defensive k-alliance inG = (V,E), k ∈ [−∆,∆] ∩ Z, if for every v ∈ S,

δS(v) ≥ δS(v) + k. (1)

We consider the value of k in the set of integers K := [−∆,∆] ∩ Z. Forsome graphs, there are some values of k ∈ K without defensive k-alliances.For instance, for k ≥ 2 do not exist defensive k-alliances in the star graph Sn

with n vertices. Besides, V (G) is a defensive δ-alliance in G if G is connected.Note that for any S with 〈S〉 connected there exists some k ∈ K such that itis a defensive k-alliance in G.

1 Email: waltercarb@gmail.com2 Email: jomaro@math.uc3m.es3 Email: jsmathguerrero@gmail.com4 Email: ytnunez@math.uc3m.es

W. Carballosa et al. / Electronic Notes in Discrete Mathematics 46 (2014) 313–320314

We denote by kS := max{k ∈ K : S is a defensive k-alliance}. We saythat kS is the exact index of alliance of S, or also, S is an exact defensivekS-alliance in G.

Remark 1.1 The exact index of alliance of S inG is kS = minv∈S

{δS(v)−δS(v)}.

We define the alliance polynomial of a graph G with variable x as fol-

lows: A(G; x) =∑

k∈K

Ak(G)xn+k, with Ak(G) the number of exact defensive

k-alliances in G. In [6] the authors introduce this alliance polynomial and [13]deals with the alliance polynomials of cubic graphs. In this paper we studythe alliance polynomials of regular graphs and their coefficients, see Section2. In Section 3 we focus on the alliance polynomials of connected regulargraphs; besides, we prove that the family of alliance polynomials of connected∆-regular graphs with small degree is a very special one, since it does notcontain alliance polynomials of graphs which are not connected ∆-regular.

2 Alliance polynomials for regular graphs

Below, a quick reminder of some previous results for general graphs (not nec-essarily regular) that will be useful, see [6]. We denote by Deg(p) the degreeof the polynomial p.

Theorem 2.1 Let G be any graph. Then A(G; x) satisfies the following prop-erties:

(i) A(G; x) does not have zeros in the interval (0,∞).

(ii) A(G; 1) < 2n, and it is the number of connected induced subgraphs in G.

(iii) If G has at least an edge and its degree sequence has exactly r differentvalues, then A(G; x) has at least r + 1 terms.

(iv) A(G; x) is a symmetric polynomial (i.e., an even or an odd function ofx) if and only if the degree sequence of G has either all values odd or alleven.

(v) A−∆(G) and A−∆+1(G) are the number of vertices in G with degree ∆and ∆− 1, respectively.

(vi) A∆(G) is equal to the number of connected components in G which are∆-regular.

(vii) n+ δ ≤ Deg(A(G; x)) ≤ n +∆.

As usual, by cycle we mean a simple closed curve, i.e., a path with different

W. Carballosa et al. / Electronic Notes in Discrete Mathematics 46 (2014) 313–320 315

vertices, unless the last one, which is equal to the first vertex. The followinglemma is a well known result of graph theory.

Lemma 2.2 If r ≥ 2 is a natural number and G is any graph with δ(v) ≥ rfor every v ∈ V (G), then there exists a cycle η in G with L(η) ≥ r + 1.

We show now some results about the alliance polynomial of regular graphsand their coefficients, for details see [7]. If G is a graph and v ∈ V (G), wedenote by G \ {v} the subgraph obtained by removing from G the vertex vand the edges incident to v. We say that v is a cut vertex if G \ {v} has moreconnected components than G. Besides, if p is a polynomial we denote byDegmin(p) the minimum degree of their non-zero coefficients.

Theorem 2.3 For any ∆-regular graph G, its alliance polynomial A(G; x)satisfies the following properties:

(i) A−∆+2i(G) is the number of connected induced subgraphs of G with min-imum degree i (0 ≤ i ≤ ∆).

(ii) Degmin

(

A(G; x))

= n−∆ and A−∆(G) = n.

(iii) Deg(

A(G; x))

= n+∆. Besides, 2n = Degmin

(

A(G; x))

+Deg(

A(G; x))

and 8m = Deg2(

A(G; x))

−Deg2min

(

A(G; x))

.

(iv) 1 ≤ A∆(G) ≤ n/(∆ + 1). Furthermore, G is a connected graph if andonly if A∆(G) = 1.

(v) If ∆ > 0, then A−∆+2(G) ≥ m and A∆−2(G) ≥ n+n0 with n0 the numberof cut vertices; in particular, A∆−2(G) ≥ n.

(vi) A(G; x) is either an even or an odd function of x. Furthermore, A(G; x)is an even function of x if and only if n +∆ is even.

(vii) The unique real zero of A(G; x) is x = 0, and its multiplicity is n−∆.

Theorem 2.4 Let G be any connected graph. Then G is regular if and onlyif A∆(G) = 1.

Proof. If G is regular, then by Theorem 2.1 vi) we obtain A∆(G) = 1. Be-sides, if A∆(G) = 1, then there is an exact defensive ∆-alliance S in G withδS(v) ≥ δS(v) + ∆ ≥ ∆ (i.e., δS(v) = ∆ and δS(v) = 0) for every v ∈ S. So,the connectivity of G gives that G is a ∆-regular graph. ✷

Theorem 2.5 Let G1, G2 be two regular graphs. If A(G1; x) = A(G2; x),then G1 and G2 have the same order, size, degree and number of connectedcomponents.

Proof. The proof is direct using Theorem 2.3 ii-iii) and Theorem 2.1 vi). ✷

W. Carballosa et al. / Electronic Notes in Discrete Mathematics 46 (2014) 313–320316

The next theorem characterizes the degree of any regular graph by thenumber of non-zero coefficients of its alliance polynomial.

Theorem 2.6 Let G be any ∆-regular graph with order n. Then A(G; x) has

∆ + 1 non-zero coefficients. Furthermore, A(G; x) =

∆∑

i=0

A∆−2i(G) xn+∆−2i,

with A−∆(G) = n, A∆(G) ≥ 1, and A∆−2i(G) ≥n(∆

i)

min{∆,n−i}for 1 ≤ i ≤

∆− 1 if ∆ > 0.

Proof (Sketch) By Theorem 2.3 we obtain the structure of A(G; x) withA−∆(G) = n and A∆(G) ≥ 1. Hence, we build an exact defensive (∆ − 2i)-alliance for 1 ≤ i ≤ ∆−1 and we obtain a lower bound of the number of exactdefensive (∆− 2i)-alliance with the given structure. For details see [7]. ✷

The following results in this section are interesting properties of the coef-ficients of A(G; x) that will be useful along this work.

Theorem 2.7 Let G be any ∆-regular graph with order n < 2∆. Then wehave A∆−2(G) = n.

Proof (Sketch) Note that G is a Hamiltonian graph with ∆ ≥ 2, hence, wehave that V \ {u} is an exact defensive (∆− 2)-alliance in G for every u ∈ Vand so A∆−2(G) ≥ n. Besides, we prove that there is not an exact defensive(∆− 2)-alliance S ⊂ V with |S| ≤ n − 2. In order to complete the proof weproceed by contradiction using the double counting technic. For details see[7]. ✷

Lemma 2.8 Let G be any ∆-regular graph with order n, ∆ ≥ 3 and 2∆ ≤n ≤ 2∆+ 1. If G contains two cliques of cardinality ∆, then these cliques aredisjoint. In particular, G contains at most two cliques of cardinality ∆.

Proof (Sketch) Seeking for a contradiction, assume that there exist cliquesS1, S2 ⊂ V of cardinality ∆ with S1∩S2 6= ∅. We compute δS1∪S2

(u) for everyu ∈ S1 ∩ S2 and we obtain |S1 ∩ S2| = ∆ − 1. In fact, ∆ = 3 and n = 6.Therefore, G is a graph isomorphic to either K3,3 or the Cartesian productP2✷K3. This finishes the proof since it is impossible to have three disjointcliques of cardinality ∆ contained in G. ✷

Remark 2.9 Let G be any ∆-regular graph with order n and ∆ ≥ 1 suchthat G has two disjoint cliques of cardinality ∆. Then:

(i) If n = 2∆, then G is isomorphic to the Cartesian product graph P2✷K∆.

W. Carballosa et al. / Electronic Notes in Discrete Mathematics 46 (2014) 313–320 317

(ii) If n = 2∆ + 1, then ∆ is even (since n∆ = 2m) and G can be obtainedfrom P2✷K∆ by removing ∆/2 copy edges of P2 and connecting the ∆vertices with degree ∆− 1 with a new vertex.

Theorem 2.10 Let G be any ∆-regular graph with order n, size m, ∆ ≥ 3and 2∆ ≤ n ≤ 2∆ + 1. Then n ≤ A∆−2(G) ≤ n+m+ 2.

Theorem 2.11 Let G be a ∆-regular connected graph with order n and letG∗ be a graph with order n1 and, minimum and maximum degrees δ1 and ∆1,respectively. If A(G∗; x) = A(G; x), then G∗ is a connected graph with exactlyn vertices of degree ∆1 = ∆+ n1 − n, n1 ≥ n, ∆1 ≥ ∆ and δ1 ≡ ∆1(mod 2).

Furthermore, if n1 > n, then ∆1+δ1+22

≤ ∆.

The proofs of Theorems 2.10 and 2.11 are long and they use some previousresults with combinatorial arguments, for details see [7].

3 Alliance polynomials of regular graphs with small de-gree

The theorems in this section can be seen as a natural continuation of [6,13]in the sense of showing the distinctive power of the alliance polynomial of agraph. In particular, we show that the family of alliance polynomials of ∆-regular graphs with small degree ∆ is a special family of alliance polynomialssince there not exists a non ∆-regular graph with alliance polynomial equalto one of their members, see Theorems 3.1 and 3.4.

Theorem 3.1 Let G be a ∆-regular graph with 0 ≤ ∆ ≤ 3 and G∗ anothergraph. If A(G∗; x) = A(G; x), then G∗ is a ∆-regular graph with the sameorder, size and number of connected components of G.

Proof (Sketch) In [6, Theorem 2.7] the authors obtain the uniqueness of thealliance polynomials of 0-regular graphs (the empty graphs). The result for1 ≤ ∆ ≤ 2 follows from [6, Theorems 3.3] and Theorems 2.1 and 2.6. Finally,[13, Theorem 2.6] gives the result for ∆ = 3. For details see [7]. ✷

Now we prove a similar result for ∆-regular graphs with ∆ > 3. First, weprove some technical results that will be useful.

Lemma 3.2 Let G1 be a graph with minimum and maximum degree δ1 and∆1, respectively, and let n ≥ 3 be a fixed natural number. Assume that G1 hasorder n1 > n with exactly n vertices of degree ∆1, and such that its alliancepolynomial A(G1; x) is symmetric. The following statements hold:

W. Carballosa et al. / Electronic Notes in Discrete Mathematics 46 (2014) 313–320318

(i) If δ1 = 1, then A(G1; x) is not a monic polynomial of degree 2n−n1+∆1.

(ii) If δ1 = 2, then we have 2n1 < 2∆1 + n or A(G1; x) is not a monicpolynomial of degree 2n− n1 +∆1.

Proof (Sketch) We proceed by contradiction, i.e., we assume that A(G1; x)is a monic polynomial with degree 2n−n1+∆1. The argument in the proof ofTheorem 2.11 gives that G1 is a connected graph with a determined structuresince there is an unique exact defensive

[

∆1−2(n1−n)]

-alliance in G1. UsingTheorem 2.11 obtain 1 ≤ δ1 ≤ 2, so, working separately in each case we obtaincontradictions; this provides the result. For details see [7]. ✷

Lemma 3.3 Let G1 be a graph with minimum and maximum degree 2 and∆1, respectively, and let n ≥ 3 be a fixed natural number. Assume that G1 hasorder n1 > n with exactly n vertices of degree ∆1, and such that its alliancepolynomial A(G1; x) is symmetric. If n < 2[∆1 − (n1 − n)] and A(G1; x) is amonic polynomial of degree 2n− n1 +∆1, then A2(n−n1)+∆1−2(G1) > n.

Proof (Sketch) As in the proof of Lemma 3.2 we have that G1 is a connectedgraph with a determined structure since there is an unique exact defensive[

∆1−2(n1−n)]

-alliance in G1. We continue in this way obtaining n+1 exactdefensive

[

∆1− 2(n1−n)− 2]

-alliances in G1, and this finishes the proof. Fordetails see [7]. ✷

Theorem 3.4 Let G be a connected ∆-regular graph with ∆ ≤ 5 and G∗

another graph. If A(G∗; x) = A(G; x), then G∗ is a connected ∆-regular graphwith the same order and size of G.

Proof. If 0 ≤ ∆ ≤ 3, then the result follows from Theorem 3.1. Assume that4 ≤ ∆ ≤ 6. Let n, n1 be the orders of G,G∗, respectively, and let δ1,∆1 bethe minimum and maximum degree of G∗, respectively. By Theorem 2.11, G∗

is a connected graph and n1 ≥ n. Seeking for a contradiction assume thatn1 > n. By Theorem 2.11 and Lemma 3.2 we can obtain a contradiction if∆ = 4. If ∆ = 5 then by Theorem 2.11 we have δ1 = 1 and ∆1 = 7, orδ1 = 2 and ∆1 = 6. Hence, by Lemma 3.2 we have directly a contradictionin the first case. In the second case, since n1 = n + 1, Lemma 3.2 gives thatn < 10. Thus, Lemma 3.3 and Theorem 2.7 gives n < A2(G

∗) = A3(G) = n.For details see [7]. ✷

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