The ABC index is one of the most applicable topological graph indices and several properties of it has been studied already due to its extensive chemical applications. Several variants of it have also been defined and used for several reasons. In this paper, we calculate the atom-bond connectivity index of some derived graphs such as double graphs, subdivision graphs and complements of some standard graphs.Publisher's Versio

Cangül, İsmail Naci

Prakasha, Kunkanadu Nanjundappa

Reddy, P. Siva Kota

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. and Eng. Math. V.13, N.2, 2023, pp. 400-408ATOM-BOND-CONNECTIVITY INDEX OF CERTAIN GRAPHSK. N. PRAKASHA1, P. S. K. REDDY2, I. N. CANGUL3∗, §Abstract. The ABC index is one of the most applicable topological graph indices andseveral properties of it has been studied already due to its extensive chemical applica-tions. Several variants of it have also been defined and used for several reasons. In thispaper, we calculate the atom-bond connectivity index of some derived graphs such asdouble graphs, subdivision graphs and complements of some standard graphs.Keywords: Atom-bond-connectivity index, double graphs, subdivision graphs, k-complement.AMS Subject Classification: 05C07, 05C30, 05C50.1. Introduction and preliminariesLet G be a graph with n vertices v1, v2, · · · , vn and m edges. In this paper, we onlyconsider simple, undirected and signless graphs.The complement of a graph G is a graph G having the same vertex set such that twodistinct vertices of G are adjacent if and only if they are not adjacent in G. The com-plement is a useful tool in many calculations, especially with the topological graph indices.Let G be a graph and Pk = {V1, V2, · · · , Vk} be a partition of its vertex set V . Simi-larly to the complement of a graph, the k-complement of G is defined as follows: For allpairs of sets Vi and Vj in Pk for i 6= j, remove the edges between Vi and Vj and add theedges between the vertices of Vi and Vj which are not in G and this k-complement will bedenoted by Gk.1 Dept. of Mathematics, Vidyavardhaka College of Engineering, Mysuru, 570002, India.e-mail: prakashamaths@gmail.com, ORCID: https://orcid.org/0000-0002-6908-4076.2 Dept. of Mathematics, Sri Jayachamarajendra College of Engineering, JSS Science and TechnologyUniversity, Mysuru, 570 006, India.e-mail: pskreddy@jssstuniv.in, ORCID: https://orcid.org/0000-0003-4033-8148.3 Dept. of Mathematics, Faculty of Arts and Science, Bursa Uludag University, 16059, Bursa, Turkey.e-mail: cangul@uludag.edu.tr, ORCID: https://orcid.org/0000-0002-0700-5774.∗ Corresponding author.§ Manuscript received: March 1, 2020; accepted: May 13, 2020.TWMS Journal of Applied and Engineering Mathematics, Vol.13, No.2 © Işık University, Departmentof Mathematics, 2023; all rights reserved.400PRAKASHA, REDDY, CANGUL: ABC INDEX OF CERTAIN GRAPHS 401Let G be a graph and again Pk be a partition of its vertex set V as above. Then thek(i)-complement of G, yet another type of the complement of a graph, is defined as fol-lows: For each partition set Vr in Pk, remove the edges of G joining the vertices in Vr andadd the edges of G (complement of G) joining the vertices of Vr, and it is denoted by Gk(i).Topological graph indices are defined and used in many areas to study several propertiesof different objects such as atoms and molecules. These indices can be classified intoseveral classes according to their way of definition: by means of matrices, by means ofvertex degrees, by means of distances, etc. One of these topological indices is the atombond-connectivity index (ABC index) which is defined in terms of the vertex degrees asfollows:ABC(G) =∑uv∈E(G)√du + dv − 2dudv.In [6], the ABC index for some graphs is calculated together with GA, Randic and Zagrebindices. In [3], the first Zagreb index and multiplicative Zagreb coindices of graphs werecalculated. There are many papers on the subject, see e.g. [1, 4].A derived graph is the graph obtained from any given graph following some predefinedrule. One of these derived graphs is the subdivision graph. The subdivision graph S(G) ofa simple graph G is defined as the new graph obtained by adding an extra vertex into eachedge of G. The subdivision graph of the xcycle graph Cn is shown in Fig. 1. The subdi-vision graphs have been studied in literature, [2], [7], [8], [9], [10] and [14]. All versions ofFigure 1. Subdivision graph of the cycle graph CnZagreb indices and coindices of subdivision graphs of certain graph types were studied in[13]. The Zagreb indices and multiplicative Zagreb indices of subdivision graphs of doublegraphs were studied in [15].In [11] and [12], a generalization of subdivision graphs were defined and studied: ther-subdivision graph of a graph G, which is denoted by Sr(G), is the new graph obtainedfrom G by replacing each of its edges by a path of length r+1; or equivalently by insertingr additional vertices into each edge of G. Clearly, in the case of r = 1, the obtained402 TWMS J. APP. AND ENG. MATH. V.13, N.2, 20231-subdivision graph is the subgraph.In [5], topological indices of the derived graphs of subdivision graphs were studied.For a graph G with vertex set V (G) = {v1, v2, · · · , vn}, we take another copy of Gwith vertices labelled by {v1, v2, · · · , vn}, this time, where vi corresponds to vi for eachi. If we connect vi to the neighbours of vi for each i, we obtain a new graph called thedouble graph of G. It is denoted by D(G). In Fig. 2 and 3, the double graphs of P6 andC4 are given: The Zagreb indices and multiplicative Zagreb indices of double graphs ofFigure 2. Double graph of P6Figure 3. Double graph of C4subdivision graphs were studied in [10].2. Atom-bond-connectivity index of double graphsSeveral topological graph indices of some of the derived graphs have already been cal-culated. In this section, we calculate the atom-bond-connectivity indices of the doublegraphs of some classes of frequently-used graphs. In the proofs, we use some countingalgorithm where each edge is named according to the vertex degrees of the edge: An edgee is called an (r, s)-edge if two end vertices of e has vertex degrees r and s. For example, alledges of a cycle graph are (2, 2)-edges and the edges of a path graph are either (1, 2)-edgesor (2, 2)-edges.PRAKASHA, REDDY, CANGUL: ABC INDEX OF CERTAIN GRAPHS 403Theorem 2.1.ABC(D(G)) =4√2 + (n− 3)√6 if G = Pn,n√6 if G = Cn,n√4n− 6 if G = Kn,2n√4n− 2 if G = Kn,n,2√2(n− 1) if G = K1,n−1.Proof. We shall give the proof for two of the graph types: For Pn and for Cn. Proofs ofthe other graph types can be done in a similar way by counting the edges and applyingthe formula.Note that the double graph D(Pn) of the path graph Pn has 4n− 4 edges. There are 8(2, 4)-edges and 4(n− 3) (4, 4)-edges. That is, the edge diagram of D(Pn) is as follows:(r, s)-edges their number(2, 4)-edges 8(4, 4)-edges 4(n− 3)ThereforeABC(D(Pn)) =∑uv∈E(D(Pn))√du+dv−2dudv= 8√2+4−22·4 + 4(n− 3)√4+4−24·4= 4√2 + (n− 3)√6.Secondly, consider D(Cn). D(Cn) has 4n (4, 4)-edges. Therefore its edge diagram issimpler:(r, s)-edges their number(4, 4)-edges 4nand using this information, we calculate the ABC index of the double graph of the cyclegraph Cn as follows:ABC(D(Cn)) =∑uv∈E(D(Cn))√du+dv−2dudv= 4n√4+4−24·4= n√6.3. Atom-bond-connectivity index of complementsIn this section, we calculate the ABC indices of several complements of some graphclasses. We first start with the ordinary complements:Theorem 3.1.ABC(G) =(n−1)√2n−62 if G = K1,n−1,0 if G = Kn×2,n√22√n− 4 if G = Cn.Proof. Let first G = K1,n−1 be the star graph of order n. Then the complement of thisgraph is K1,n−1 = Kn−1, that is, the complete graph of order n− 1. The edge diagram ofthis complement graph is404 TWMS J. APP. AND ENG. MATH. V.13, N.2, 2023(r, s)-edges their number(n− 2, n− 2)-edges (n−1)(n−2)2This implies that the ABC index of this complement graph isABC(K1,n−1) =∑uv∈E(K1,n−1)√du+dv−2dudv= (n−1)(n−2)2√n−2+n−2−2(n−2)·(n−2)= (n−1)√2n−62 .Secondly, consider the complement of the coctail party graph Kn×2. It is not difficultto see that this complement is the ladder rung graph LRn consisting of n P2’s. The edgediagram of this complement graph is(r, s)-edges their number(1, 1)-edges nThis implies that the ABC index of this complement graph isABC(K1,n−1) =∑uv∈E(K1,n−1)√du+dv−2dudv= (n−1)(n−2)2√n−2+n−2−2(n−2)·(n−2)= (n−1)√2n−62 .Finally, consider the complement of the cycle graph Cn. It is clear that in the complement,every vertex is connected to all the vertices except three: the vertex itself and its twoneigbours. Therefore the edge diagram of this complement graph would be(r, s)-edges their number(n− 3, n− 3)-edges n(n−3)2This implies that the ABC index of the complement graph ABC(Cn) isABC(Cn) =∑uv∈E(Cn)√du+dv−2dudv= n(n−3)2√n−3+n−3−2(n−3)·(n−3)= n2√2n− 8giving the result. The following results on other types of complements of some graphs can similarly beproven by constructing the edge diagram and using the index formula:Theorem 3.2.ABC(K(n×2)2) =5n− 6n√2n− 2.Theorem 3.3.ABC((Sn,n)2i) = (2n− 2)√3n− 52n2 − 4n + 2+ (2n2 − 5n + 3)√4n− 62n− 2.PRAKASHA, REDDY, CANGUL: ABC INDEX OF CERTAIN GRAPHS 4054. Atom-bond-connectivity index of subdivision graphsSubdivision graphs are important tools in calculating several properties of larger graphsin terms of smaller graphs. Now we calculate the atom-bond-connectivity index of thesubdivision graphs of some frequently-used graphs. First we give a useful result for regulargraphs:Lemma 4.1. Let G be an r-regular graph of arder n and size m. Then the following aresatisfied:(i) ABC(G) = mr√2r − 2;(ii) ABC(S(G)) = m√2.Proof. Both items can be proven by the definition. Theorem 4.1.ABC(S(G)) =√2(n− 1) if G = Pn,√2n if G = Cn,n2−n√2if G = Kn,√2(n− 1) if G = K1,n−1Proof. We shall give the proof for two of the graph types: For Pn and for K1,n−1. Proofsof the other graph types can be done in a similar way by counting the edges and applyingthe formula.First, we consider the subdivision graph of the path graph Pn. It is known that thisgraph is S(Pn) = P2n−1. The edge diagram of this complement graph would be(r, s)-edges their number(1, 2)-edges 2(2, 2)-edges 2n− 4.This implies that the ABC index of the complement graph ABC(Cn) isABC(S(Pn)) = ABC(P2n−1)=∑uv∈E(S(Pn))√du+dv−2dudv= 2√1+2−21·2 + (2n− 4)√2+2−22·2=√2(n− 1)..Secondly, we consider the subdivision graph of the star graph K1,n−1. It is known thatthis graph is the double star graph. The edge diagram of this complement graph wouldbe(r, s)-edges their number(1, 2)-edges n− 1(2, n− 1)-edges n− 1.406 TWMS J. APP. AND ENG. MATH. V.13, N.2, 2023This implies that the ABC index of the complement graph ABC(S(K1,n−1)) isABC(S(K1,n−1)) =∑uv∈E(S(K1,n−1))√du+dv−2dudv= (n− 1)√1+2−21·2 + (n− 1)√2+n−1−22·(n−1)=√2(n− 1).5. Atom-bond-connectivity index of the friendship graph, its subdivisiongraph and its line graphTheorem 5.1.ABC(G) =3n√2if G = F 3n ,6n√2if G = S(F 3n),√2n + (2n−1)32√2if G = L(F 3n),Proof. We shall prove only the last item as the other two are similar to the above proofs.Line graph of a graph is an important tool as one of the derived graphs. Let L(F 3n) bethe line graph of the friendship graph which is the graph consisting of n triangles whichall have a common vertex. Then the line graph of it is a graph obtained by adding ntriangles around a complete graph on 2n vertices. It has the following edge diagram:(r, s)-edges their number(2, 2n)-edges 2n(2n, 2n)-edges 2n(2n−1)2.This implies that the ABC index of the complement graph ABC(L(F 3n)) isABC(L(F 3n)) =∑uv∈E(L(F 3n))√du+dv−2dudv= 2n√2+2n−22·2n +2n(2n−1)2√2n+2n−22n·2n=√2n + (2n−1)32√2.Finally we consider the ABC index of another interesting graph. First we recall thedefinition:Definition 5.1. Jahangir graphs Jn,m for m = 3, is a graph on nm + 1 vertices. i.e.,it is a graph consisting of a cycle Cnm with one additional vertex which is adjacent to mvertices of Cnm at distance n to each other on Cnm.Theorem 5.2.ABC(J2,m) =√2m + m√m+13mProof. Similar to the above. PRAKASHA, REDDY, CANGUL: ABC INDEX OF CERTAIN GRAPHS 4076. Results and discussionThe fact that modelling a molecule by a graph gives us many required information onthe physico-chemical properties of the molecule at the end of some mathematical calcula-tions made on the graph has been used in the last seven decades. For this, we calculatethe values of the so called topological graph index and also determine which values couldbe attained by this index.Derived graphs are graphs produced from a given graph according to some rule. Therelations between a graph and its derived graph help us to obtain information on one fromthe other. In this paper, three of the important derived graphs are studied and their atom-bond-connectivity indices are determined. The methods and calculations given here canbe generalized to other graph thypes and also to other derived graphs. It is also possibleto determine the ABC index of the graph operations.Disclosure statementNo potential conflict of interest was reported by the authors.References[1] Aytac, V., Turaci, T., (2018), On arithmetic-geometric index (GA) and edge GA index, TWMS J.App. Eng. Math., 8 (1), pp. 61-70.[2] Bhanavari, S., Devanaboina, S., Kuncham, S. P., (2017), A note on line graphs, TWMS J. App. Eng.Math., 7 (1), pp. 173-179.[3] Das, Kinkar Ch., Akgunes, N., Togan, M., Yurttas, A., Cangul, I. N., Cevik, A. S., (2016), On thefirst Zagreb index and multiplicative Zagreb coindices of graphs, Analele Stiintifice ale UniversitatiiOvidius Constanta, 24 (1), pp. 153-176.[4] Ediz, S., (2018), Maximal graphs of the first reverse Zagreb beta index, TWMS J. App. Eng. Math.,8 (1a), pp. 306-310.[5] Hosamani, S. M., Lokesha, V., Cangul, I. N., Devendraiah, K. M., (2016), On Certain TopologicalIndices of the Derived Graphs of Subdivision Graphs, TWMS J. App. Eng. Math., 6 (2), pp. 324-332.[6] Lokesha, V., Shetty, S. B., Ranjini, P. S., Cangul, I. N., (2015), Computing ABC, GA, Randic andZagreb Indices, Enlightments of Pure and Applied Mathematics, 1 (1), pp. 17-28.[7] Mahde, S. S., Mathad, V., (2016), Hub integrity of splitting graph and duplication of graph elements,TWMS J. App. Eng. Math., 6 (2), pp. 289-297.[8] Ranjini, P. S., Lokesha, V., Cangul, I. N., (2011), On the Zagreb indices of the line graphs of thesubdivision graphs, Appl. Math. Comput., 218, pp. 699-702.[9] Ranjini, P. S., Rajan, M. A., Lokesha, V., (2010), On Zagreb Indices of the Subdivision Graphs, Int.J. of Math. Sci. & Eng. Appns (IJMSEA), 4, pp. 221-228.[10] Togan, M., Yurttas, A., Cevik, A. S., Cangul, I. N., (2019), Zagreb indices and multiplicative Zagrebindices of double graphs of subdivision graphs, TWMS, J. App. Eng. Math., 9 (2), pp. 404-412.[11] Togan, M., Yurttas, A., Cangul, I. N., (2017), Zagreb and multiplicative Zagreb indices of r-subdivisiongraphs of double graphs, Scientia Magna, 12 (1), pp. 115-119.[12] Togan, M., Yurttas, A., Cangul, I. N., (2015), Some formulae and inequalities on several Zagrebindices of r-subdivision graphs, Enlightments of Pure and Applied Mathematics, 1 (1), pp. 29-45.[13] Togan, M., Yurttas, A., Cangul, I. N., (2016), All versions of Zagreb indices and coindices of subdi-vision graphs of certain graph types, Advanced Studies in Contemporary Mathematics, 26 (1), pp.227-236.[14] Vukicevic, D., Gutman, I., Furtula, B., Andova, V., Dimitrov, D., (2011), Some Observations onComparing Zagreb Indices, MATCH Commun. Math. Comput. Chem., 66, 627-645.[15] Yurttas, A., Togan, M., Cangul, I. N., (2016), Zagreb indices and multiplicative Zagreb indices ofsubdivision graphs of double graphs, Advanced Studies in Contemporary Mathematics, 26 (3), 407-416.408 TWMS J. APP. AND ENG. MATH. V.13, N.2, 2023Prof. K. N. Prakasha for the photography and short autobiography, see TWMS J. App. and Eng.Math. V.9, N.4.P. Siva Kota Reddy received his M.Sc. and Ph.D. degrees from University ofMysore and his D.Sc. in Mathematics from Berhampur University, India. Currently,he is working as a professor of Mathematics at Sri Jayachamarajendra College ofEngineering, JSS Science and Technology University.Ismail Naci Cangul for the photography and short autobiography, see TWMS J. App. and Eng.Math. V.6, N.2.

Atom-bond-connectivity index of certain graphs

http://acikerisim.isikun.edu.tr/xmlui/bitstream/11729/5469/1/ATOM_BOND_CONNECTIVITY_INDEX_OF_CERTAIN_GRAPHS.pdf

Atom-bond-connectivity index of certain graphs

Abstract

Similar works

Full text

Available Versions

Isik University Academic Open Access