EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 5, 2020, 1149-1161 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Special Issue Dedicated to Professor Hari M. Srivastava On the Occasion of his 80th Birthday Computing Discrete Adriatic Indices of Probabilistic Neural Network T. Deepika1,∗, V. Lokesha2 1 Department of Mathematics, Dayananda Sagar University, Bengaluru, Karnataka, India. 2 Department of Mathematics, Vijayanagara Sri Krishnadevaraya University, Ballari, Karnataka, India. Abstract. A Topological index is a numeric quantity which characterizes the whole structure of a graph. Adriatic indices are also part of topological indices, mainly it is classified into two namely extended variables and discrete adriatic indices, especially, discrete adriatic indices are analyzed on the testing sets provided by the International Academy of Mathematical Chemistry (IAMC) and it has been shown that they have good presaging substances in many compacts. This contrived attention to compute some discrete adriatic indices of probabilistic neural network. 2020 Mathematics Subject Classifications: 05C09,05C30 Key Words and Phrases: Topological indices, probabilistic neural network, degree vertex. 1. Prologue and Provocation In this work peculiar attention is compensated to Adriatic indices that have been de- fined by D. Vukičević and M. Gasperov. Discrete Adriatic indices are the family of 148 bond-additive topological indices defined as follows. Adr(G) = ∑ uv∈E(G) γj(ϕi,a(tu), ϕi,a(tv)) where the variables and functions can take the following values: ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3712 Email addresses: v.lokesha@gmail.com (V. Lokesha), sastry.deepi@gmail.com (T. Deepika) https://www.ejpam.com 1149 c© 2020 EJPAM All rights reserved. T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1150 tu ∈ {du, Du} i ∈ {1, 2, 3} j ∈ {1, 2, ....., 8} a =  ( −1,−1 2 , 1 2 , 1, 2 ) if i = 2 and j ∈ (1, 2, ...5)( 1 2 , 1, 2 ) otherwise and i ∈ (1, 2) (12 , 2) if i = 3 ϕ1,a(x) = loga(x)(a > 0); ϕ2,a(x) = xa(a ∈ R \ 0); ϕ3,a(x) = ax(a > 0); Now, the naming convention for the discrete adriatic indices is as follows: 1. γ1 corresponds to Randi type; γ1(x, y) = x.y • Randi type lodeg index: Adr(G) = ∑ uv∈E(G) γ1(ϕ1,1(du), ϕ1,1(dv)) • Randi type sdi index: Adr(G) = ∑ uv∈E(G) γ1(ϕ2,2(Du), ϕ2,2(Dv)) • Randi type hadi index: Adr(G) = ∑ uv∈E(G) γ1(ϕ3,1/2(Du), ϕ3,1/2(Dv)) 2. γ2 corresponds to sum; γ2(x, y) = x+ y • sum lordeg index: Adr(G) = ∑ uv∈E(G) γ2(ϕ1,1/2(du), ϕ1,1/2(dv)) 3. γ3 corresponds to inverse sum; γ3(x, y) =  1 x+ y , if x+ y 6= 0 0, otherwise. • inverse sum lordeg index: Adr(G) = ∑ uv∈E(G) γ3(ϕ1,1/2(du), ϕ1,1/2(dv)) • inverse sum indeg index: Adr(G) = ∑ uv∈E(G) γ3(ϕ2,−1(du), ϕ2,−1(dv)) 4. γ4 corresponds to misbalance; γ4(x, y) =| x− y | • misbalance lodeg index: Adr(G) = ∑ uv∈E(G) γ4(ϕ1,1(du), ϕ1,1(dv)) T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1151 • misbalance losdeg index: Adr(G) = ∑ uv∈E(G) γ4(ϕ1,2(du), ϕ1,2(dv)) • misbalance rodeg index: Adr(G) = ∑ uv∈E(G) γ4(ϕ2,1/2(du), ϕ2,1/2(dv)) • misbalance deg index: Adr(G) = ∑ uv∈E(G) γ4(ϕ2,1(du), ϕ2,1(dv)) • misbalance hadeg index: Adr(G) = ∑ uv∈E(G) γ4(ϕ3,1/2(du), ϕ3,1/2(dv)) 5. γ5 corresponds to inverse misbalance; γ5(x, y) =  1 | x− y | , if x 6= y 0, if x = y • misbalance indeg index: Adr(G) = ∑ uv∈E(G) γ5(ϕ2,−1(du), ϕ2,−1(dv)) • misbalance irdeg index: Adr(G) = ∑ uv∈E(G) γ5(ϕ2,−1/2(du), ϕ2,−1/2(dv)) • misbalance indi index: Adr(G) = ∑ uv∈E(G) γ5(ϕ2,−1(Du), ϕ2,−1(Dv)) 6. γ6 corresponds to min-max; γ6(x, y) =  min{x, y} max{x, y} , if max{x, y} 6= 0 0, if max{x, y} = 0 • min-max rodeg index: Adr(G) = ∑ uv∈E(G) γ6(ϕ2,1/2(du), ϕ2,1/2(dv)) • min-max sdi index: Adr(G) = ∑ uv∈E(G) γ6(ϕ2,2(Du), ϕ2,2(Dv)) 7. γ7 corresponds to max-min; γ7(x, y) =  max{x, y} min{x, y} , if min{x, y} 6= 0 0, if min{x, y} = 0 • max-min rodeg index: Adr(G) = ∑ uv∈E(G) γ7(ϕ2,1/2(du), ϕ2,1/2(dv)) T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1152 • max-min deg index: Adr(G) = ∑ uv∈E(G) γ7(ϕ2,1(du), ϕ2,1(dv)) • max-min sdeg index: Adr(G) = ∑ uv∈E(G) γ7(ϕ2,2(du), ϕ2,2(dv)) 8. γ8 corresponds to symmetric division; γ8(x, y) =  x y + y x , if x, y 6= 0 0, otherwise • symmetric division deg index: Adr(G) = ∑ uv∈E(G) γ8(ϕ2,1(du), ϕ2,1(dv)) The Probabilistic Neural Network (PNN) gives new directions in the Quantitative Structure-Activity Relationship (QSAR)/Quantitative Structure-Property Relationship (QSPR) studies [3]. The PNN methodology is applicable to classification problems and the basic underlying theory behind these probability-based methods is presented along with applications of the PNN methodology. The PNN model presented identifies molecules as potential soluble epoxide hydrolase in- hibitors using a binary classification scheme. This network inputs consist of a small set of descriptors that encode structural features at the molecular level (refer [4, 14]). The topological indices are exploited to hypothesis the physical features related to the bio-activities, chemical reactivities in certain networks. In this article, we established the degree-based discrete adriatic indices of the probabilistic neural network. Recently, [1, 12] exposed the 148 adriatic indices among which Symmetric division deg (SDD) index is one of the discrete adriatic indices, and it has been proved as a good predictor for total surface area for polychlorobiphenyls. Newly, Lokesha et. al., [5, 6, 8] worked out for SDD index of unicyclic and bicyclic later it is also extended to tricyclic and tetracyclic, also they did for graph operations. Newly, Deepika et al [9], computed the graph structure of 2D-Lattice, nanotube and nan- otorus exploiting the definitions of SDD and other topological indices via certain graph operators which gives more attraction towards to this index (also see [2, 10, 13]). In the last decade the probabilistic neural networks are widely studied in different classification problems. These are applied to solve the problems related to email security enhancement and in the intrusion detection systems. The probabilistic neural networks are also applied in medicine such as for detecting resistivity to antibiotics, for diagnosing hep- atitis, and for the quantification and segmentation of brain tissues from MR images (also see [7, 11]). In the present study, we compute the topological indices such as newly defined adriatic indices to continue the progressive study of the probabilistic neural networks. T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1153 The construction of the probabilistic neural network. we consider the probabilistic neural network consisting on three layers of nodes. • The first layer called by input layer has a certain number of nodes, the second layer called by hidden layer consists on a certain number of classes such that each class contains a particular number of nodes, and the third layer called by output layer has a number of nodes equal to the number of classes of the second layer. • The architecture of a probabilistic neural network, each node of input layer is con- nected to all the nodes of each class of the hidden layer and all the nodes of each class of the hidden layer are connected to a unique node of the output layer. • Assume the input layer has n nodes, the hidden layer consists on k classes such that each class has m nodes, and the third layer called by output layer has k nodes. Thus, a probabilistic neural network denoted by PNN(n, k,m) is such that |V (PNN(n, k,m))| = v = n+ k(m+ 1) and |E(PNN(n, k,m))| = e = km(n+ 1), where (n, k,m) ∈ N (set of natural numbers). In Fig. 1, the probabilistic neural network is shown for n = 4, k = 2 and m = 3. Figure 1: The probabilistic neural network PNN(4, 2, 3) We define the partitions of the edge set of PNN(n, k,m) with respect to degree of ver- tices. There are two types of edges with respect to degrees of end vertices in PNN(n, k,m), namely with degrees of end vertices km, n+ 1 and degrees of end vertices n+ 1,m. Thus, we have shown in the following table 1. Table 1: The edge partition of the edges of PNN(n, k,m) based on degrees of end vertices E{d(u),d(v)} Ekm,n+1 En+1,m E{d(u),d(v)} nkm km 2. Main Results In this section, we established some results for probabilistic neural networks using different adriatic indices T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1154 Theorem 1. Let G be the probabilistic neural network PNN(n, k,m) then for (n, k,m) ≥ 1, its misbalance deg index, misbalance indeg index, misbalance rodeg index and misbalance irdeg index is given by MBIα(G) =  nkm|km− (n+ 1)|+ km|(n+ 1)−m| for α = 1 nkm| 1 km − 1 n+ 1 |+ km| 1 n+ 1 − 1 m | for α = −1 nkm| √ km− √ n+ 1|+ km| √ n+ 1− √ m| for α = 1 2 nkm| 1√ km − 1√ n+ 1 |+ km| 1√ n+ 1 − 1√ m | for α = −1 2 Proof. Let G be the PNN(n, k,m), for (n, k,m) ≥ 1. The number of vertices and edges in PNN(n, k,m) are n+k(m+1) and km(n+1) respectively. There are two types of edges in PNN(n, k,m) based on degrees of end vertices of each edge. Table 1 shows such an edge partition of PNN(n, k,m). Now by using the definition of Misbalance indices and table 1 we obtain the required results as follows We consider the following cases for the possible values of α Case 1 : α = 1 Applying the formula of Misbalance deg index for α = 1 and by using edge partition given in table 1 we get MBI1 = ∑ uv∈E(G) | du − dv | = ∑ uv∈Ekm,n+1 | du − dv | + ∑ uv∈En+1,m | du − dv | = nkm | km− (n+ 1) | +km | (n+ 1)−m | = km [ n | km− (n+ 1) | + | (n+ 1)−m | ] Case 2 : α = −1, Applying the formula of Misbalance indeg index for α = −1 and by using edge partition given in table 1 we get MBI−1 = ∑ uv∈E(G) | 1 du − 1 dv | = ∑ uv∈Ekm,n+1 | 1 du − 1 dv | + ∑ uv∈En+1,m | 1 du − 1 dv | = nkm | 1 km − 1 n+ 1 | +km | 1 n+ 1 − 1 m | = km [ n | 1 km − 1 n+ 1 | + | 1 n+ 1 − 1 m | ] T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1155 Case 3 : α = 1 2 , Edge partition for Misbalance rodeg index whenα = 1 2 we get MBI 1 2 = ∑ uv∈E(G) | √ du − √ dv | = ∑ uv∈Ekm,n+1 | √ du − √ dv | + ∑ uv∈En+1,m | √ du − √ dv | = nkm | √ km− √ n+ 1 | +km | √ n+ 1− √ m | = [ | √ km− √ n+ 1 | + | √ n+ 1− √ m | ] Case 4 : α = −1 2 , For Misbalance irdeg index when α = −1 2 using edge partition given in table 1 we get MBI− 1 2 = ∑ uv∈E(G) | 1√ du − 1√ dv | = ∑ uv∈Ekm,n+1 | 1√ du − 1√ dv | + ∑ uv∈En+1,m | 1√ du − 1√ dv | = nkm | 1√ km − 1√ n+ 1 | +km | 1√ n+ 1 − 1√ m | = km [ n | 1√ km − 1√ n+ 1 | + | 1√ n+ 1 − 1√ m | ] Theorem 2. Let G be the probabilistic neural network PNN(n, k,m) then for (n, k,m) ≥ 1, its max-min deg index, max-min sdeg index and max rodeg index is given by Max-min deg index = k n+ 1 (m2nk + (n+ 1)2) Max-min sdeg index = k [ k2m4n+ (n+ 1)4 (n+ 1)2m ] Max-rodeg index = km [ n √ km n+ 1 + √ n+ 1 m ] Proof. Let G be the PNN(n, k,m), for (n, k,m) ≥ 1. The number of vertices and edges in PNN(n, k,m) are n+k(m+1) and km(n+1) respectively. There are two types of edges in PNN(n, k,m) based on degrees of end vertices of each edge. Table 1 shows such an edge partition of PNN(n, k,m). Now by using the formulas of max-min deg index, max-min sdeg index, max rodeg index and table 1 we obtain the required results as follows Max-min deg index = ∑ uv∈E(G) max(du, dv) min(du, dv) T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1156 = ∑ uv∈Ekm,n+1 max(du, dv) min(du, dv) + ∑ uv∈En+1,m max(du, dv) min(du, dv) = nkm ( km n+ 1 ) + km ( n+ 1 m ) = k n+ 1 (m2nk + (n+ 1)2) Max-min sdeg index = ∑ uv∈E(G) ( max(du, dv) min(du, dv) )2 = ∑ uv∈Ekm,n+1 ( max(du, dv) min(du, dv) )2 + ∑ uv∈En+1,m ( max(du, dv) min(du, dv) )2 = nkm ( km n+ 1 )2 + km ( n+ 1 m )2 = k [ k2m4n+ (n+ 1)4 (n+ 1)2m ] Max rodeg index = ∑ uv∈E(G) √ max(du, dv) min(du, dv) = ∑ uv∈Ekm,n+1 √ max(du, dv) min(du, dv) + ∑ uv∈En+1,m √ max(du, dv) min(du, dv) = nkm √ km n+ 1 + km √ n+ 1 m = km [ n √ km n+ 1 + √ n+ 1 m ] Theorem 3. Let G be the probabilistic neural network PNN(n, k,m) then for (n, k,m) ≥ 1, its min-max rodeg index and min-max sdi index is given by Min-max rodeg index = km [ n √ n+ 1 km + √ m n+ 1 ] Min-max sdi index = n(n+ 1)4 + k2m4 km(n+ 1)2 Proof. Let G be the PNN(n, k,m), for (n, k,m) ≥ 1. The number of vertices and edges in PNN(n, k,m) are n+ k(m+ 1) and km(n+ 1) respectively. There are two types of edges in PNN(n, k,m) based on degrees of end vertices of each edge. Table 1 shows T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1157 such an edge partition of PNN(n, k,m). Now by using the formulas of min-max rodeg index, min-max sdi index and table 1 we obtain the required results as follows Min-max rodeg index = ∑ uv∈E(G) √ min(du, dv) max(du, dv) = ∑ uv∈Ekm,n+1 √ min(du, dv) max(du, dv) + ∑ uv∈En+1,m √ min(du, dv) max(du, dv) = nkm √ n+ 1 km + km √ m n+ 1 = km [ n √ n+ 1 km + √ m n+ 1 ] Min-max sdi index = ∑ uv∈E(G) ( min(du, dv) max(du, dv) )2 = ∑ uv∈Ekm,n+1 ( min(du, dv) max(du, dv) )2 + ∑ uv∈En+1,m ( min(du, dv) max(du, dv) )2 = nkm ( n+ 1 km )2 + km ( m n+ 1 )2 = n(n+ 1)4 + k2m4 km(n+ 1)2 Theorem 4. Let G be the probabilistic neural network PNN(n, k,m) then for (n, k,m) ≥ 1, its Inverse sum indeg index and Symmetric division deg index is given by ISI(G) = km2 [ k(m+ n)(2n+ 1) + (n+ 1)2 (m+ n+ 1)(km+ n+ 1) ] SDD(G) = 1 n+ 1 [ n(n+ 1)2 + k(m2(1 + kn) + n2 + 3) ] Proof. Let G be the PNN(n, k,m), for (n, k,m) ≥ 1. The number of vertices and edges in PNN(n, k,m) are n+ k(m+ 1) and km(n+ 1) respectively. There are two types of edges in PNN(n, k,m) based on degrees of end vertices of each edge. Table 1 shows such an edge partition of PNN(n, k,m). Now by using the formulas of ISI, SDD and table 1 we obtain the required results as follows ISI(G) = ∑ uv∈E(G) dudv du + dv T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1158 = ∑ uv∈Ekm,n+1 dudv du + dv + ∑ uv∈En+1,m dudv du + dv = nkm [ (km)(n+ 1) km+ (n+ 1) ] + km [ (n+ 1)(m) (n+ 1) +m = km [ n ( km km+ 1 ) + ( m 1 +m )] = km2 [ k(m+ n)(2n+ 1) + (n+ 1)2 (m+ n+ 1)(km+ n+ 1) ] Min−maxsdiindex = ∑ uv∈E(G) ( min(du, dv) max(du, dv) )2 = ∑ uv∈Ekm,n+1 ( min(du, dv) max(du, dv) )2 + ∑ uv∈En+1,m ( min(du, dv) max(du, dv) )2 = nkm ( n+ 1 km )2 + km ( m n+ 1 )2 = n(n+ 1)4 + k2m4 km(n+ 1)2 Let G be the probabilistic neural network PNN(n, k,m) then for (n, k,m) ≥ 1, its adriatic indices of PNN of γj(x, y) (j = 1, 2, ...8) index is given by • Adr(PNN)γ1(x,y) =  km.log(n+ 1).log(knmn+1), if ϕi,a = ϕ1,1 2n4 + 58k4 + 6m4 + 60, if ϕ2,2 2 22n + 58 22k + 6 22m + 15, if ϕ3,1/2. • Adr(PNN)γ2(x,y) = km(n+ 1) √ log(n+ 1) + √ log(km) + √ log(m), if ϕi,a = ϕ1,1/2 • Adr(PNN)γ3(x,y) =  1 km(n+ 1) √ log(n+ 1) + √ log(km) + √ log(m) , if ϕi,a = ϕ1,1/2 km2 [ nk(n+ 1) km+ n+ 1 + n+ 1 m+ n+ 1 ] , if ϕ2,−1 • Adr(PNN)γ4(x,y) =  km[log(knmn−1)− (n− 1)log(n+ 1)], if ϕi,a = ϕ1,1 2km[log(knmn−1)− (n− 1)log(n+ 1)], if ϕ1,2 nkm | √ km− √ n+ 1 | +km | √ n+ 1− √ m |, if ϕ2,1/2 nkm | km− (n+ 1) | +km | (n+ 1)−m |, if ϕ2,1 km[n2−km + 2n+1 − n2−(n+1) − 2−m], if ϕ3,1/2 T. Deepika, V. Lokesha / Eur. J. Pure Appl. Math, 13 (5) (2020), 1149-1161 1159 • Adr(PNN)γ5(x,y) =  nkm | 1 km − 1 n+1 | +km | 1 n+1 − 1 m |, if ϕi,a = ϕ2,−1 nkm | 1√ km − 1√ n+1 | +km | 1√ n+1 − 1√ m |, if ϕ2,−1/2 0, if ϕ2,−1 • Adr(PNN)γ6(x,y) = km [ n √ n+1 km + √ m n+1 ] , if ϕi,a = ϕ2,1/2 14(n+ k +m), if ϕ2,2 • Adr(PNN)γ7(x,y) =  km [ n √ km n+1 + √ n+1 m ] , if ϕi,a = ϕ2,1/2 k n+1(m2nk + (n+ 1)2), if ϕ2,1 k [ k2m4n+(n+1)4 (n+1)2m ] , if ϕ2,2 • Adr(PNN)γ8(x,y) = 1 n+1 [ n(n+ 1)2 + k(m2(1 + kn) + n2 + 3) ] , if ϕi,a = ϕ2,1 Figure 2: The probabilistic neural network PNN(n, n, 1) 3. Conclusion We compare the results of the computed adriatic indices for the probabilistic neural network PNN(n, k,m) with the help of software package. In the main results the formulae of all the indices are computed in term of n, k, and m, where n is number of nodes in first layer, the second layer is consisting on k classes such that each class has m nodes and the third layer contains k nodes. Moreover, the total number of vertices in the probabilistic neural network PNN(n, k,m) (order of PNN(n, k,m)) is |V (PNN(n, k,m))| = v = n + k(m + 1). If we assume k = n and m = 1, then the probabilistic neural network becomes PNN(n, n, 1) with order v = 3n, where n is a natural number. In Fig. 2, along the horizontal line the values of v for the probabilistic neural network REFERENCES 1160 PNN(n, n, 1) are taken and along the vertical line the computed values of the indices are shown. Among them the Randic type index is dominant. Moreover, all the adriatic indices remain constant approximately with increasing values of v. In this paper, the adriatic indices of the probabilistic neural network are studied and the analytical closed formulas are determined that will help to understand the underlying topologies related to the physical features of this network. Acknowledgements The authors thank the referees for their useful comments that helped to improve this paper. References [1] V. Alexander. Upper and lower bounds of symmetric division deg index. Iranian Journal of Mathematical Chemistry, 5(2):91–98, 2014. [2] N. Ananda, P.S. Ranjini, V. Lokesha, and S.A. Wazzan. Subdivision and semitotal point graphs of Archimedean lattices on some toplogical indices. volume 22, pages 830–837. Proceedings of the Jangieon Mathematical society, 2019. [3] J. Devillers and A. T. Balaban (Eds.). Gordon and Breach, Amsterdam, 1999. [4] Budak F.U and Beyli E.D. Detection of resistivity for antibiotics by probabilistic neural networks. Journal of Medical Systems, 35:87–91, 2011. [5] C. K. Gupta, V. Lokesha, and S. B. Shetty. 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