Let G be a graph with vertex set V=(v1,v2,…,vn). Let δ(vi) be the degree of the vertex vi∈V. If the vertices vi1,vi2,…,vih+1 form a path of length h≥1 in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1/δ(vi1)δ(vi2)⋯δ(vih+1) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for Rh, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.

1. Introduction

In this work we consider simple graphs G=(V,E) with n vertices and m edges. Let V=(v1,v2,…,vn) be the vertex set of G. For every vertex vi∈V, δ(vi) represents the degree of the vertex vi in G. The maximum and minimum degree of the vertices of G will be denoted by Δ and δ, respectively.

The Randić index R1(G) of a graph G was introduced in 1975 [1] and defined as R1(G)=∑vivj∈E1δ(vi)δ(vj).
This graph invariant, sometimes referred to as connectivity index, has been successfully related to a variety of physical, chemical, and pharmacological properties of organic molecules, and it has became into one of the most popular molecular-structure descriptors. After the publication of the first paper [1], mathematical properties of R1 were extensively studied, see [2–6] and the references cited therein.

The higher-order Randić indices are also of interest in chemical graph theory. For h≥1, the hth order Randić index Rh(G) of a graph G is defined as Rh(G)=∑vi1vi2⋯vih+1∈Ph(G)1δ(vi1)δ(vi2)⋯δ(vih+1),
where 𝒫h(G) denotes the set of paths of length h contained (as subgraphs) in G. Of the higher-order Randić indices the most frequently applied is R2 [7–10]. Estimations of the higher-order Randić index of regular graphs and semiregular bipartite graphs are given in [10]. In this paper we are interested in studying the higher-order Randić index, Rh, for corona product graphs. Roughly speaking, we study the cases h=1, h=2 for arbitrary graphs and the case h≥3 when the second factor of the corona product is an empty graph. As an example of a chemical compound whose graph is obtained as a corona product graph we consider the Cycloalkanes with a single ring, whose chemical formula is CkH2k, and whose molecular graph can be expressed as Ck⊙N2, where Ck is the cycle graph of order k and N2 is the empty graph of order two. We recall that, given two graphs G and H of order n1 and n2, respectively, the corona product G⊙H is defined as the graph obtained from G and H by taking one copy of G and n1 copies of H and then joining by an edge each vertex of the ith copy of H with the ith vertex of G.

2. Estimating <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M63"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> for Corona GraphsTheorem 2.1.

For i∈{1,2}, let Gi be a graph of minimum degree δi, maximum degree Δi, order ni and size mi. Then,
R1(G1⊙G2)≤m1δ1+n2+n1m2δ2+1+n1n2(δ1+n2)(δ2+1),R1(G1⊙G2)≥m1Δ1+n2+n1m2Δ2+1+n1n2(Δ1+n2)(Δ2+1).

Proof.

Let Gi=(Vi,Ei), i∈{1,2}, and let G1⊙G2=(V,E). We have
R1(G1⊙G2)=∑xy∈E1δ(x)δ(y)=Q1+Q2+Q3,
where
Q1=∑ab∈E11(δ(a)+n2)(δ(b)+n2)≥m1Δ1+n2,Q2=∑uv∈E21(δ(u)+1)(δ(v)+1)≥n1m2Δ2+1,Q3=∑a∈V1,u∈V21(δ(a)+n2)(δ(u)+1)≥n1n2(Δ1+n2)(Δ2+1).
Thus, the lower bound follows. Analogously we deduce the upper bound.

Corollary 2.2.

For i∈{1,2}, let Gi be a δi-regular graph of order ni. Then,
R1(G1⊙G2)=n1δ12(δ1+n2)+n1n2δ22(δ2+1)+n1n2(δ1+n2)(δ2+1).

Theorem 2.3.

For i∈{1,2}, let Gi be a graph of minimum degree δi, maximum degree Δi, order ni, and size mi. Then,
R2(G1⊙G2)≤n1(δ2+1)δ1+n2(n2(n2-1)2+2m2)+1δ1+n2(2n2m1δ2+1+∑δ(vi)≥2δ(vi)(δ(vi)-1)2δ(vi)+n2)+12(δ2+1)∑δ(ui)≥2δ(ui)(δ(ui)-1)δ(ui)+1,R2(G1⊙G2)≥n1(Δ2+1)Δ1+n2(n2(n2-1)2+2m2)+1Δ1+n2(2n2m1Δ2+1+∑δ(vi)≥2δ(vi)(δ(vi)-1)2δ(vi)+n2)+12(Δ2+1)∑δ(ui)≥2δ(ui)(δ(ui)-1)δ(ui)+1.

Proof.

Let V1={v1,v2,…,vn1} and V2={u1,u2,…,un2} be the set of vertices of G1 and G2, respectively. Given a vertex v∈Vi, we denote by NGi(v) the set of neighbors that v has in Gi. The paths of length two in G1⊙G2 are obtained as follows:

paths uivjuk, i≠k, where ui,uk∈V2 and vj∈V1,

paths uivjvk, j≠k, where ui∈V2 and vjvk∈V1,

paths viujuk, j≠k, where vi∈V1 and uj,uk∈V2,

paths of length two belonging to G1,

paths of length two belonging to the n1 copies of G2.

So, we have R2(G1⊙G2)=∑i=15Qi, where
Q1=∑vj∈V1;ui,uk∈V21(δ(ui)+1)(δ(vj)+n2)(δ(uk)+1)=∑j=1n11δ(vj)+n2⋅∑i=1n2-1∑l=i+1n21(δ(ui)+1)(δ(ul)+1)≥n1n2(n2-1)2(Δ2+1)Δ1+n2
corresponds to the paths type (i),
Q2=∑ui∈V2;vj,vk∈V11(δ(ui)+1)(δ(vj)+n2)(δ(vk)+n2)=∑i=1n21δ(ui)+1⋅∑j=1n1∑vl∈NG1(vj)1(δ(vj)+n2)(δ(vl)+n2)≥2m1n2(Δ1+n2)Δ2+1
corresponds to the paths type (ii),
Q3=∑vi∈V1;uj,uk∈V21(δ(vi)+n2)(δ(uj)+1)(δ(uk)+1)=∑i=1n11δ(vi)+n2⋅∑j=1n2∑ul∈NG2(uj)1(δ(uj)+1)(δ(ul)+1)≥2n1m2(Δ2+1)Δ1+n2
corresponds to the paths type (iii),
Q4=∑vivjvk∈P(G1)1(δ(vi)+n2)(δ(vj)+n2)(δ(vk)+n2)≥12(Δ1+n2)∑δ(vi)≥2δ(vi)(δ(vi)-1)δ(vi)+n2
corresponds to the paths type (iv), and
Q5=∑uiujuk∈P(G2)1(δ(ui)+1)(δ(uj)+1)(δ(uk)+1)≥12(Δ2+1)∑δ(ui)≥2δ(ui)(δ(ui)-1)δ(ui)+1
corresponds to the paths type (v). Thus, the lower bound follows. The upper bound is obtained by analogy.

Corollary 2.4.

For i∈{1,2}, let Gi be a δi-regular graph of order ni. Then,
R2(G1⊙G2)=n1n2(δ2+1)δ1+n2(n2-12+δ2)+n1δ12(δ1+n2)(2n2δ2+1+δ1-1δ1+n2)+n2δ2(δ2-1)2(δ2+1)δ2+1.

The girth of a graph is the size of its smallest cycle. For instance, the molecular graphs of benzenoid hydrocarbons have girth 6. The molecular graphs of biphenylene and azulene have girth 4 and 5, respectively [11].

The following result, and its proof, was implicitly obtained in the proof of Theorem 1 of [10]. By completeness, here we present it as a separate result.

Lemma 2.5.

Let G=(V,E) be a graph with girth g(G). If δ≥2 and g(G)>h, then the number of paths of length h in G is bounded by
(δ-1)h-22∑u∈Vδ(u)(δ(u)-1)≤|Ph(G)|≤(Δ-1)h-22∑u∈Vδ(u)(δ(u)-1).

Proof.

Since δ≥2, for every v∈V, the number of paths of length 2 in G of the form vivvj is δ(v)(δ(v)-1)/2. Therefore, the result follows for h=2.

Suppose now that h≥3. Given a vertex u∈V, let 𝒫h(u) be the set of paths of length h whose second vertex is u, that is, paths of the form u1uu2⋯uh. We denote by N(v) the set of neighbors of an arbitrary vertex v∈V. Note that the degree of v is δ(v)=|N(v)|. If δ≥2, then for every v∈V and w∈N(v) we have N(w)∖{v}≠∅. So, for every u∈V, there exists a vertex sequence u1uu2⋯uh such that u1,u2∈N(u), u3∈N(u2)∖{u}, u4∈N(u3)∖{u2},…,anduh∈N(uh-1)∖{uh-2}. If g(G)>h, then the sequence u1uu2⋯uh is a path. Conversely, every path of length h whose second vertex is u can be constructed as above. Hence, the number of paths of length h whose second vertex is u is bounded by
|Ph(u)|≥minu1uu2⋯uh∈Ph(u){δ(u)(δ(u)-1)∏j=2h-1(δ(uj)-1)}≥δ(u)(δ(u)-1)(δ-1)h-2,|Ph(u)|≤maxu1uu2⋯uh∈Ph(u){δ(u)(δ(u)-1)∏j=2h-1(δ(uj)-1)}≤δ(u)(δ(u)-1)(Δ-1)h-2.
Thus, the result follows.

Now Nk denotes the empty graph of order k.

Theorem 2.6.

Let G=(V,E) be a graph with girth g(G), minimum degree δ, and maximum degree Δ. If δ≥2 and g(G)>h≥3, then
Rh(G⊙Nk)≤(Δ-12δ+k+k)(Δ-1)h-3(δ+k)h/2∑u∈Vδ(u)(δ(u)-1),Rh(G⊙Nk)≥(δ-12Δ+k+k)(δ-1)h-3(Δ+k)h/2∑u∈Vδ(u)(δ(u)-1).

Proof.

The paths of length h in G contribute to Rh(G⊙Nk) in
∑vi1vi2⋯vih+1∈Ph(G)1∏l=1h+1(δ(vil)+k).
Moreover, each path of length h-1 in G leads to 2k paths of length h in G⊙Nk; thus, the paths of length h-1 in G contribute to Rh(G⊙Nk) in
∑vi1vi2⋯vih∈Ph-1(G)2k∏l=1h(δ(vil)+k).
Hence,
Rh(G⊙Nk)=∑vi1vi2⋯vih+1∈Ph(G)1∏l=1h+1(δ(vil)+k)+∑vi1vi2⋯vih∈Ph-1(G)2k∏l=1h(δ(vil)+k)≤|Ph(G)|(δ+k)h+1+2k|Ph-1(G)|(δ+k)h.
By Lemma 2.5 we obtain the upper bound and the lower bound is obtained by analogy.

Corollary 2.7.

Let G=(V,E) be a δ-regular graph of order n and girth g(G). If δ≥2 and g(G)>h≥3, then
Rh(G⊙Nk)=(δ-12δ+k+k)nδ(δ-1)h-2(δ+k)h/2.

Acknowledgment

This work was partly supported by the Spanish Government through projects TSI2007-65406-C03-01 “E-AEGIS” and CONSOLIDER INGENIO 2010 CSD2007-00004 “ARES.”

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