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ISRN Discrete Mathematics
VolumeΒ 2011Β (2011), Article IDΒ 262183, 7 pages
http://dx.doi.org/10.5402/2011/262183
Research Article

On the Randić Index of Corona Product Graphs

Departamento d'Enginyeria InformΓ tica i MatemΓ tiques, Universitat Rovira i Virgili, Avinguda PaΓ―sos Catalans 26, 43007 Tarragona, Spain

Received 24 July 2011; Accepted 20 September 2011

Academic Editor: X.Β Yong

Copyright Β© 2011 Ismael G. Yero and Juan A. RodrΓ­guez-VelΓ‘zquez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐺 be a graph with vertex set 𝑉=(𝑣1,𝑣2,…,𝑣𝑛). Let 𝛿(𝑣𝑖) be the degree of the vertex π‘£π‘–βˆˆπ‘‰. If the vertices 𝑣𝑖1,𝑣𝑖2,…,π‘£π‘–β„Ž+1 form a path of length β„Žβ‰₯1 in the graph 𝐺, then the β„Žth order RandiΔ‡ index π‘…β„Ž of 𝐺 is defined as the sum of the terms 1/𝛿(𝑣𝑖1)𝛿(𝑣𝑖2)⋯𝛿(π‘£π‘–β„Ž+1) over all paths of length β„Ž contained (as subgraphs) in 𝐺. Lower and upper bounds for π‘…β„Ž, 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 π‘š edges. Let 𝑉=(𝑣1,𝑣2,…,𝑣𝑛) be the vertex set of 𝐺. For every vertex π‘£π‘–βˆˆπ‘‰, 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 Ξ” and 𝛿, respectively.

The RandiΔ‡ index 𝑅1(𝐺) of a graph 𝐺 was introduced in 1975 [1] and defined as 𝑅1(𝐺)=π‘£π‘–π‘£π‘—βˆˆπΈ1𝛿𝑣𝑖𝛿𝑣𝑗.(1.1) 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 RandiΔ‡ indices are also of interest in chemical graph theory. For β„Žβ‰₯1, the β„Žth order RandiΔ‡ index π‘…β„Ž(𝐺) of a graph 𝐺 is defined as π‘…β„Žξ“(𝐺)=𝑣𝑖1𝑣𝑖2β‹―π‘£π‘–β„Ž+1βˆˆπ’«β„Ž(𝐺)1𝛿𝑣𝑖1𝛿𝑣𝑖2ξ€Έξ‚€π‘£β‹―π›Ώπ‘–β„Ž+1,(1.2) where π’«β„Ž(𝐺) denotes the set of paths of length β„Ž contained (as subgraphs) in 𝐺. Of the higher-order RandiΔ‡ indices the most frequently applied is 𝑅2 [7–10]. Estimations of the higher-order RandiΔ‡ index of regular graphs and semiregular bipartite graphs are given in [10]. In this paper we are interested in studying the higher-order RandiΔ‡ index, π‘…β„Ž, for corona product graphs. Roughly speaking, we study the cases β„Ž=1, β„Ž=2 for arbitrary graphs and the case β„Žβ‰₯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 πΆπ‘˜π»2π‘˜, and whose molecular graph can be expressed as πΆπ‘˜βŠ™π‘2, where πΆπ‘˜ is the cycle graph of order π‘˜ 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 πΊβŠ™π» 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 π‘…β„Ž for Corona Graphs

Theorem 2.1. For π‘–βˆˆ{1,2}, let 𝐺𝑖 be a graph of minimum degree 𝛿𝑖, maximum degree Δ𝑖, order 𝑛𝑖 and size π‘šπ‘–. Then, 𝑅1𝐺1βŠ™πΊ2ξ€Έβ‰€π‘š1𝛿1+𝑛2+𝑛1π‘š2𝛿2+𝑛+11𝑛2𝛿1+𝑛2𝛿2ξ€Έ,𝑅+11𝐺1βŠ™πΊ2ξ€Έβ‰₯π‘š1Ξ”1+𝑛2+𝑛1π‘š2Ξ”2+𝑛+11𝑛2Δ1+𝑛2Ξ”ξ€Έξ€·2ξ€Έ.+1(2.1)

Proof. Let 𝐺𝑖=(𝑉𝑖,𝐸𝑖), π‘–βˆˆ{1,2}, and let 𝐺1βŠ™πΊ2=(𝑉,𝐸). We have 𝑅1𝐺1βŠ™πΊ2ξ€Έ=π‘₯π‘¦βˆˆπΈ1βˆšπ›Ώ(π‘₯)𝛿(𝑦)=𝑄1+𝑄2+𝑄3,(2.2) where 𝑄1=ξ“π‘Žπ‘βˆˆπΈ11𝛿(π‘Ž)+𝑛2𝛿(𝑏)+𝑛2ξ€Έβ‰₯π‘š1Ξ”1+𝑛2,𝑄2=ξ“π‘’π‘£βˆˆπΈ21√β‰₯𝑛(𝛿(𝑒)+1)(𝛿(𝑣)+1)1π‘š2Ξ”2,𝑄+13=ξ“π‘Žβˆˆπ‘‰1,π‘’βˆˆπ‘‰21𝛿(π‘Ž)+𝑛2ξ€Έβ‰₯𝑛(𝛿(𝑒)+1)1𝑛2Δ1+𝑛2Ξ”ξ€Έξ€·2ξ€Έ.+1(2.3) Thus, the lower bound follows. Analogously we deduce the upper bound.

Corollary 2.2. For π‘–βˆˆ{1,2}, let 𝐺𝑖 be a 𝛿𝑖-regular graph of order 𝑛𝑖. Then, 𝑅1𝐺1βŠ™πΊ2ξ€Έ=𝑛1𝛿12𝛿1+𝑛2ξ€Έ+𝑛1𝑛2𝛿22𝛿2ξ€Έ+𝑛+11𝑛2𝛿1+𝑛2𝛿2ξ€Έ+1.(2.4)

Theorem 2.3. For π‘–βˆˆ{1,2}, let 𝐺𝑖 be a graph of minimum degree 𝛿𝑖, maximum degree Δ𝑖, order 𝑛𝑖, and size π‘šπ‘–. Then, 𝑅2𝐺1βŠ™πΊ2≀𝑛1𝛿2ξ€Έβˆš+1𝛿1+𝑛2𝑛2𝑛2ξ€Έβˆ’12+2π‘š2ξƒͺ+1𝛿1+𝑛2βŽ›βŽœβŽœβŽœβŽ2𝑛2π‘š1βˆšπ›Ώ2++1𝛿𝑣𝑖β‰₯2π›Ώξ€·π‘£π‘–π›Ώξ€·π‘£ξ€Έξ€·π‘–ξ€Έξ€Έβˆ’12𝛿𝑣𝑖+𝑛2⎞⎟⎟⎟⎠+12𝛿2+1𝛿𝑒𝑖β‰₯2π›Ώξ€·π‘’π‘–π›Ώξ€·π‘’ξ€Έξ€·π‘–ξ€Έξ€Έβˆ’1𝛿𝑒𝑖,𝑅+12𝐺1βŠ™πΊ2ξ€Έβ‰₯𝑛1ξ€·Ξ”2ξ€Έβˆš+1Ξ”1+𝑛2𝑛2𝑛2ξ€Έβˆ’12+2π‘š2ξƒͺ+1Ξ”1+𝑛2βŽ›βŽœβŽœβŽœβŽ2𝑛2π‘š1βˆšΞ”2++1𝛿𝑣𝑖β‰₯2π›Ώξ€·π‘£π‘–π›Ώξ€·π‘£ξ€Έξ€·π‘–ξ€Έξ€Έβˆ’12𝛿𝑣𝑖+𝑛2⎞⎟⎟⎟⎠+12ξ€·Ξ”2+1𝛿𝑒𝑖β‰₯2π›Ώξ€·π‘’π‘–π›Ώξ€·π‘’ξ€Έξ€·π‘–ξ€Έξ€Έβˆ’1𝛿𝑒𝑖.+1(2.5)

Proof. Let 𝑉1={𝑣1,𝑣2,…,𝑣𝑛1} and 𝑉2={𝑒1,𝑒2,…,𝑒𝑛2} be the set of vertices of 𝐺1 and 𝐺2, respectively. Given a vertex π‘£βˆˆπ‘‰π‘–, we denote by 𝑁𝐺𝑖(𝑣) the set of neighbors that 𝑣 has in 𝐺𝑖. The paths of length two in 𝐺1βŠ™πΊ2 are obtained as follows: (i)paths π‘’π‘–π‘£π‘—π‘’π‘˜, π‘–β‰ π‘˜, where 𝑒𝑖,π‘’π‘˜βˆˆπ‘‰2 and π‘£π‘—βˆˆπ‘‰1, (ii)paths π‘’π‘–π‘£π‘—π‘£π‘˜, π‘—β‰ π‘˜, where π‘’π‘–βˆˆπ‘‰2 and π‘£π‘—π‘£π‘˜βˆˆπ‘‰1, (iii)paths π‘£π‘–π‘’π‘—π‘’π‘˜, π‘—β‰ π‘˜, where π‘£π‘–βˆˆπ‘‰1 and 𝑒𝑗,π‘’π‘˜βˆˆπ‘‰2, (iv)paths of length two belonging to 𝐺1, (v)paths of length two belonging to the 𝑛1 copies of 𝐺2.
So, we have 𝑅2(𝐺1βŠ™πΊ2βˆ‘)=5𝑖=1𝑄𝑖, where 𝑄1=ξ“π‘£π‘—βˆˆπ‘‰1;𝑒𝑖,π‘’π‘˜βˆˆπ‘‰21𝛿𝑒𝑖𝛿𝑣+1𝑗+𝑛2π›Ώξ€·π‘’ξ€Έξ€·π‘˜ξ€Έξ€Έ=+1𝑛1𝑗=11𝛿𝑣𝑗+𝑛2⋅𝑛2βˆ’1𝑛𝑖=12𝑙=𝑖+11𝛿𝑒𝑖𝛿𝑒+1𝑙β‰₯𝑛+11𝑛2𝑛2ξ€Έβˆ’12ξ€·Ξ”2ξ€Έβˆš+1Ξ”1+𝑛2(2.6) corresponds to the paths type (i), 𝑄2=ξ“π‘’π‘–βˆˆπ‘‰2;𝑣𝑗,π‘£π‘˜βˆˆπ‘‰11𝛿𝑒𝑖𝛿𝑣+1𝑗+𝑛2π›Ώξ€·π‘£ξ€Έξ€·π‘˜ξ€Έ+𝑛2ξ€Έ=𝑛2𝑖=11𝛿𝑒𝑖⋅+1𝑛1𝑗=1ξ“π‘£π‘™βˆˆπ‘πΊ1(𝑣𝑗)1𝛿𝑣𝑗+𝑛2𝛿𝑣𝑙+𝑛2ξ€Έβ‰₯2π‘š1𝑛2ξ€·Ξ”1+𝑛2ξ€ΈβˆšΞ”2+1(2.7) corresponds to the paths type (ii), 𝑄3=ξ“π‘£π‘–βˆˆπ‘‰1;𝑒𝑗,π‘’π‘˜βˆˆπ‘‰21𝛿𝑣𝑖+𝑛2𝛿𝑒𝑗𝛿𝑒+1ξ€Έξ€·π‘˜ξ€Έξ€Έ=+1𝑛1𝑖=11𝛿𝑣𝑖+𝑛2⋅𝑛2𝑗=1ξ“π‘’π‘™βˆˆπ‘πΊ2(𝑒𝑗)1𝛿𝑒𝑗𝛿𝑒+1𝑙β‰₯+12𝑛1π‘š2ξ€·Ξ”2ξ€Έβˆš+1Ξ”1+𝑛2(2.8) corresponds to the paths type (iii), 𝑄4=ξ“π‘£π‘–π‘£π‘—π‘£π‘˜ξ€·πΊβˆˆπ’«1ξ€Έ1𝛿𝑣𝑖+𝑛2𝛿𝑣𝑗+𝑛2π›Ώξ€·π‘£ξ€Έξ€·π‘˜ξ€Έ+𝑛2ξ€Έβ‰₯12ξ€·Ξ”1+𝑛2𝛿𝑣𝑖β‰₯2π›Ώξ€·π‘£π‘–π›Ώξ€·π‘£ξ€Έξ€·π‘–ξ€Έξ€Έβˆ’1𝛿𝑣𝑖+𝑛2(2.9) corresponds to the paths type (iv), and 𝑄5=ξ“π‘’π‘–π‘’π‘—π‘’π‘˜ξ€·πΊβˆˆπ’«2ξ€Έ1𝛿𝑒𝑖𝛿𝑒+1𝑗𝛿𝑒+1ξ€Έξ€·π‘˜ξ€Έξ€Έβ‰₯1+12ξ€·Ξ”2+1𝛿𝑒𝑖β‰₯2π›Ώξ€·π‘’π‘–π›Ώξ€·π‘’ξ€Έξ€·π‘–ξ€Έξ€Έβˆ’1𝛿𝑒𝑖+1(2.10) corresponds to the paths type (v). Thus, the lower bound follows. The upper bound is obtained by analogy.

Corollary 2.4. For π‘–βˆˆ{1,2}, let 𝐺𝑖 be a 𝛿𝑖-regular graph of order 𝑛𝑖. Then, 𝑅2𝐺1βŠ™πΊ2ξ€Έ=𝑛1𝑛2𝛿2ξ€Έβˆš+1𝛿1+𝑛2𝑛2βˆ’12+𝛿2ξ‚Ά+𝑛1𝛿12𝛿1+𝑛22𝑛2βˆšπ›Ώ2+𝛿+11βˆ’1βˆšπ›Ώ1+𝑛2ξƒͺ+𝑛2𝛿2𝛿2ξ€Έβˆ’12𝛿2ξ€Έβˆš+1𝛿2.+1(2.11)

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 𝛿β‰₯2 and 𝑔(𝐺)>β„Ž, then the number of paths of length β„Ž in 𝐺 is bounded by (π›Ώβˆ’1)β„Žβˆ’22ξ“π‘’βˆˆπ‘‰||𝒫𝛿(𝑒)(𝛿(𝑒)βˆ’1)β‰€β„Ž||≀(𝐺)(Ξ”βˆ’1)β„Žβˆ’22ξ“π‘’βˆˆπ‘‰π›Ώ(𝑒)(𝛿(𝑒)βˆ’1).(2.12)

Proof. Since 𝛿β‰₯2, for every π‘£βˆˆπ‘‰, the number of paths of length 2 in 𝐺 of the form 𝑣𝑖𝑣𝑣𝑗 is 𝛿(𝑣)(𝛿(𝑣)βˆ’1)/2. Therefore, the result follows for β„Ž=2.
Suppose now that β„Žβ‰₯3. Given a vertex π‘’βˆˆπ‘‰, let π’«β„Ž(𝑒) be the set of paths of length β„Ž whose second vertex is 𝑒, that is, paths of the form 𝑒1𝑒𝑒2β‹―π‘’β„Ž. We denote by 𝑁(𝑣) the set of neighbors of an arbitrary vertex π‘£βˆˆπ‘‰. Note that the degree of 𝑣 is 𝛿(𝑣)=|𝑁(𝑣)|. If 𝛿β‰₯2, then for every π‘£βˆˆπ‘‰ and π‘€βˆˆπ‘(𝑣) we have 𝑁(𝑀)⧡{𝑣}β‰ βˆ…. So, for every π‘’βˆˆπ‘‰, there exists a vertex sequence 𝑒1𝑒𝑒2β‹―π‘’β„Ž such that 𝑒1,𝑒2βˆˆπ‘(𝑒), 𝑒3βˆˆπ‘(𝑒2)⧡{𝑒}, 𝑒4βˆˆπ‘(𝑒3)⧡{𝑒2},…,andπ‘’β„Žβˆˆπ‘(π‘’β„Žβˆ’1)⧡{π‘’β„Žβˆ’2}. If 𝑔(𝐺)>β„Ž, then the sequence 𝑒1𝑒𝑒2β‹―π‘’β„Ž is a path. Conversely, every path of length β„Ž whose second vertex is 𝑒 can be constructed as above. Hence, the number of paths of length β„Ž whose second vertex is 𝑒 is bounded by ||π’«β„Ž(||𝑒)β‰₯min𝑒1𝑒𝑒2β‹―π‘’β„Žβˆˆπ’«β„Ž(𝑒)𝛿(𝑒)(𝛿(𝑒)βˆ’1)β„Žβˆ’1𝑗=2ξ€·π›Ώξ€·π‘’π‘—ξ€Έξ€Έξƒ°βˆ’1β‰₯𝛿(𝑒)(𝛿(𝑒)βˆ’1)(π›Ώβˆ’1)β„Žβˆ’2,||π’«β„Ž||(𝑒)≀max𝑒1𝑒𝑒2β‹―π‘’β„Žβˆˆπ’«β„Ž(𝑒)𝛿(𝑒)(𝛿(𝑒)βˆ’1)β„Žβˆ’1𝑗=2ξ€·π›Ώξ€·π‘’π‘—ξ€Έξ€Έξƒ°βˆ’1≀𝛿(𝑒)(𝛿(𝑒)βˆ’1)(Ξ”βˆ’1)β„Žβˆ’2.(2.13) Thus, the result follows.

Now π‘π‘˜ denotes the empty graph of order π‘˜.

Theorem 2.6. Let 𝐺=(𝑉,𝐸) be a graph with girth 𝑔(𝐺), minimum degree 𝛿, and maximum degree Ξ”. If 𝛿β‰₯2 and 𝑔(𝐺)>β„Žβ‰₯3, then π‘…β„Žξ€·πΊβŠ™π‘π‘˜ξ€Έβ‰€ξƒ©Ξ”βˆ’12√ξƒͺ𝛿+π‘˜+π‘˜(Ξ”βˆ’1)β„Žβˆ’3(𝛿+π‘˜)β„Ž/2ξ“π‘’βˆˆπ‘‰π‘…π›Ώ(𝑒)(𝛿(𝑒)βˆ’1),β„Žξ€·πΊβŠ™π‘π‘˜ξ€Έβ‰₯ξƒ©π›Ώβˆ’12√ξƒͺΞ”+π‘˜+π‘˜(π›Ώβˆ’1)β„Žβˆ’3(Ξ”+π‘˜)β„Ž/2ξ“π‘’βˆˆπ‘‰π›Ώ(𝑒)(𝛿(𝑒)βˆ’1).(2.14)

Proof. The paths of length β„Ž in 𝐺 contribute to Rβ„Ž(πΊβŠ™π‘π‘˜) in 𝑣𝑖1𝑣𝑖2β‹―π‘£π‘–β„Ž+1βˆˆπ’«β„Ž(𝐺)1ξ”βˆβ„Ž+1𝑙=1𝛿𝑣𝑖𝑙+π‘˜.(2.15) Moreover, each path of length β„Žβˆ’1 in 𝐺 leads to 2π‘˜ paths of length β„Ž in πΊβŠ™π‘π‘˜; thus, the paths of length β„Žβˆ’1 in 𝐺 contribute to π‘…β„Ž(πΊβŠ™π‘π‘˜) in 𝑣𝑖1𝑣𝑖2β‹―π‘£π‘–β„Žβˆˆπ’«β„Žβˆ’1(𝐺)2π‘˜ξ”βˆβ„Žπ‘™=1𝛿𝑣𝑖𝑙+π‘˜.(2.16) Hence, π‘…β„Žξ€·πΊβŠ™π‘π‘˜ξ€Έ=𝑣𝑖1𝑣𝑖2β‹―π‘£π‘–β„Ž+1βˆˆπ’«β„Ž(𝐺)1ξ”βˆβ„Ž+1𝑙=1𝛿𝑣𝑖𝑙++π‘˜π‘£π‘–1𝑣𝑖2β‹―π‘£π‘–β„Žβˆˆπ’«β„Žβˆ’1(𝐺)2π‘˜ξ”βˆβ„Žπ‘™=1𝛿𝑣𝑖𝑙≀||𝒫+π‘˜β„Ž(||𝐺)√(𝛿+π‘˜)β„Ž+1||𝒫+2π‘˜β„Žβˆ’1(||𝐺)√(𝛿+π‘˜)β„Ž.(2.17) 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 𝛿β‰₯2 and 𝑔(𝐺)>β„Žβ‰₯3, then π‘…β„Žξ€·πΊβŠ™π‘π‘˜ξ€Έ=ξƒ©π›Ώβˆ’12√ξƒͺ𝛿+π‘˜+π‘˜π‘›π›Ώ(π›Ώβˆ’1)β„Žβˆ’2(𝛿+π‘˜)β„Ž/2.(2.18)

Acknowledgment

This work was partly supported by the Spanish Government through projects TSI2007-65406-C03-01 β€œE-AEGIS” and CONSOLIDER INGENIO 2010 CSD2007-00004 β€œARES.”

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