EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 2, 2021, 340-350 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bounds for the Topological indices of ℘ graph M. Manjunath 1,∗,V. Lokesha1, Suvarna1, Sushmitha Jain1 1 Department of Studies in Mathematics, V. S. K. University, Vinayaka Nagara, Ballari, India Abstract. Topological indices are mathematical measure which correlates to the chemical struc- tures of any simple finite graph. These are used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR). In this paper, we define op- erator graph namely, ℘ graph and structured properties. Also, establish the lower and upper bounds for few topological indices namely, Inverse sum indeg index, Geometric-Arithmetic index, Atom-bond connectivity index, first zagreb index and first reformulated Zagreb index of ℘-graph. 2020 Mathematics Subject Classifications: 05C90, 05C12 Key Words and Phrases: Jump graph, corona product and Topological index. Dedicated to Professor H. M. Srivastava on the occasion of his 80th Birth Anniversary 1. Introduction The topological index thought came out in 1947 from work done by Harold Wiener while he was working on boiling point of paraffin and he named this index as path number, now it is renamed as Wiener index [3]. Since then many other topological indices have been defined and studied. Topological indices are numerical invariants that are associated with the topological characterization of a compound. Topological indices are closely related to the toxicological, physicochemical, pharmacological properties of a chemical compound. The significance of topological indices is mainly related to their utilizing in Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relation- ship (QSPR). Now we recall some well known topological indices. In 2010, Vukicevic and Gasperov [8, 10] introduced bond-additive topological index namely, inverse sum indeg index as a significant predictor of total surface area of octane isomers and is defined as ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.3715 Email addresses: manju3479@gmail.com (M. Manjunath), v.lokesha@gmail.com (V. Lokesha), suvarnasalimath17@gmail.com (Suvarna), sushmithajain9@gmail.com (S. Jain) http://www.ejpam.com 340 c© 2021 EJPAM All rights reserved. M. Manjunath , V. Lokesha, Suvarna, S. Jain / Eur. J. Pure Appl. Math, 14 (2) (2021), 340-350 341 ISI[G] = ∑ uv∈E[G] [ dudv du+dv ] D. Vukicevic and B. Furtula (2009) [9] introduced the Geometric-Arithmetic index and is defined as GA[G] = ∑ uv∈E[G] [ 2 √ dudv du+dv ] E. Estrada and et al, (1990) [1] introduced the atom bond connectivity index and is defined as ABC[G] = ∑ uv∈E[G] [√ du+dv−2 dudv ] It display an excellent correlation with the heat formation of alkane. Gutman. I and Trinajstic. N (1972) [2] introduced the first zagreb index and is defined as M1[G] = ∑ uv∈E[G][du + dv] A. Milicevic et al. (2004) [7] introduced the first reformulated zagreb index and is defined as EM1[G] = ∑ uv∈E[G][du + dv − 2]2 Definition 1.1. [4] The jump-graph J(G) of a graph G is the graph defined on E(G) where two vertices are adjacent if and only if their corresponding edges are not adjacent in G. Definition 1.2. [6] The corona product of G�H of these two graphs is obtained by taking one copy of G and n1 copies of H and by joining each vertex of the ith copy of H to the ith vertex of G, where 1 ≤ i ≤ n1. In order to study bounds on ℘-graph, we divide the paper into few sections. The section one contains preliminaries, definitions of well known topological indices which are useful to prove our main results. Section two deals with new class of operator graph with their properties. Section three consisting of results related to bounds for defined class of graph using recalled topological indices. Paper conclude with the conclusion and references. Let G and H be graphs with vertex sets V (G), V (H) and edge sets E(G), E(H) respectively. The degree of vertex v is the number of vertices adjacent to v. Let {V (G) ⋂ V (H) = ∅|g ∈ V (G), h ∈ V (H)}. The number of vertices and number of edges in the graphs G and H are represented by n1 , n2 and m1 , m2 respectively. We have M. Manjunath , V. Lokesha, Suvarna, S. Jain / Eur. J. Pure Appl. Math, 14 (2) (2021), 340-350 342 ∆G ≥ degG(g), δG ≤ degG(g) ∆H ≥ degH(h), δH ≤ degH(h) The bounds for different topological indices are obtained by many researchers [5]. Now we will define a new class of operator graph namely J-vertex corona product of graph (℘-graph). Definition 1.3. The J(G)�H = ℘ is a graph obtained from one copy of graph J(G) and m1 copies of H and joining a vertex of V [J(G)], that is, on the ith position in J(G) to every vertex in the ith copy of H. Properties of ℘-graph: The ℘-graph has (i). m1 +m1n2 vertices (ii). m1[m2 + n2] + m1[m1−1] 2 − ∑ uv∈E(G) [ degGu+degGv−2 2 ] edges. (iii). The degree of a vertex v ∈ V [℘] is given by deg℘u = { degHu+ 1, if u ∈ V [H] degJ(G)u+ n2, ifu ∈ V [J [G]]. 2. Bounds on various topological indices of ℘ graph In this section, we formulate the bounds on the ISI, GA, ABC, M1 and EM1 indices of ℘-graph. Theorem 1. Let G and H are two simple connected graphs, then the bounds for the inverse sum indeg index of ℘ are given by ISI[℘] ≥ m1m2(∆H + 1) 2 +m1n2 [ (∆H + 1).[(m1 − 1)− 2[∆G − 1] + n2] ∆H +m1 − 2[∆G − 1] + n2 ] + [ m1(m1 − 1) 2 −m1(∆G − 1) ][ (m1 − 1)− 2[∆G − 1] + n2 2 ] and ISI[℘] ≤ m1m2(δH + 1) 2 +m1n2 [ (δH + 1).[(m1 − 1)− 2[δG − 1] + n2] δH +m1 − 2[δG − 1] + n2 ] + [ m1(m1 − 1) 2 −m1(δG − 1) ][ (m1 − 1)− 2[δG − 1] + n2 2 ] . Proof. Consider, M. Manjunath , V. Lokesha, Suvarna, S. Jain / Eur. J. Pure Appl. Math, 14 (2) (2021), 340-350 343 ISI[℘] = m1 ∑ uv∈E(H) [ (degHu+ 1).(degHv + 1) (degHu+ 1) + (degHv + 1) ] + ∑ ef∈E(J [G]) [ (degJ [G]e+ n2).(degJ [G]f + n2) (degJ [G]e+ n2) + (degJ [G]f + n2) ] + ∑ e∈V (J [G]) ∑ u∈V (H) [ (degHu+ 1).(degJ(G)e+ n2) (degHu+ 1) + (degJ(G)e+ n2) ] = m1m2 [ (degHu+ 1).(degHv + 1) (degHu+ 1) + (degHv + 1) ] +m1n2 [ (degHu+ 1).(degJ [G]e+ n2) (degHu+ 1) + (degJ [G]e+ n2) ] + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]][ (degJ [G]e+ n2).(degJ [G]f + n2) (degJ [G]e+ n2) + (degJ [G]f + n2) ] = m1m2 [ (degHu+ 1).(degHv + 1) (degHu+ 1) + (degHv + 1) ] + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]] [ [(m1 − 1)− [degGu+ degGv − 2] + n2].[(m1 − 1)− [degGu+ degGv − 2] + n2] [(m1 − 1)− [degGu+ degGv − 2] + n2] + [(m1 − 1)− [degGu+ degGv − 2] + n2] ] +m1n2 [ (degHu+ 1).[(m1 − 1)− [degGu+ degGv − 2] + n2] (degHu+ 1) + [(m1 − 1)− [degGu+ degGv − 2] + n2] ] ≥ m1m2 [ (∆H + 1).(∆H + 1) (∆H + 1) + (∆H + 1) ] +m1n2 [ (∆H + 1).[(m1 − 1)− [∆G + ∆G − 2] + n2] (∆H + 1) + [(m1 − 1)− [∆G + ∆G − 2] + n2] ] + [[ m1(m1 − 1) 2 ] −m1 [ ∆G + ∆G − 2 2 ]] [ [(m1 − 1)− [∆G + ∆G − 2] + n2].[(m1 − 1)− [∆G + ∆G − 2] + n2] [(m1 − 1)− [∆G + ∆G − 2] + n2] + [(m1 − 1)− [∆G + ∆G − 2] + n2] ] ≥ m1m2(∆H + 1) 2 +m1n2 [ (∆H + 1).[(m1 − 1)− 2[∆G − 1] + n2] ∆H + 1 +m1 − 1− 2[∆G − 1] + n2 ] + [[ m1(m1 − 1) 2 ] −m1 [ 2(∆G − 1) 2 ]][ (m1 − 1)− 2[∆G − 1] + n2 2 ] . ISI[℘] ≥ m1m2(∆H + 1) 2 +m1n2 [ (∆H + 1).[(m1 − 1)− 2[∆G − 1] + n2] ∆H +m1 − 2[∆G − 1] + n2 ] + [ m1(m1 − 1) 2 −m1(∆G − 1) ][ (m1 − 1)− 2[∆G − 1] + n2 2 ] . Similarly, ISI[℘] ≤ m1m2(δH + 1) 2 +m1n2 [ (δH + 1).[(m1 − 1)− 2[δG − 1] + n2] δH +m1 − 2[δG − 1] + n2 ] M. Manjunath , V. Lokesha, Suvarna, S. Jain / Eur. J. Pure Appl. Math, 14 (2) (2021), 340-350 344 + [ m1(m1 − 1) 2 −m1(δG − 1) ][ (m1 − 1)− 2[δG − 1] + n2 2 ] . Theorem 2. Let G and H are two simple connected graphs, then the bounds for the geometric-arithmetic index of ℘ are given by GA[℘] ≥ m1m2 + 2m1n2 √ (∆H + 1).[(m1 − 1)− 2[∆G − 1] + n2] ∆H +m1 − 2[∆G − 1] + n2 + m1(m1 − 1) 2 −m1(∆G − 1) and GA[℘] ≤ m1m2 + 2m1n2 √ (δH + 1).[(m1 − 1)− 2[δG − 1] + n2] δH +m1 − 2[δG − 1] + n2 + m1(m1 − 1) 2 −m1(δG − 1). Proof. Consider GA[℘] = m1 ∑ uv∈E(H) [ 2 √ (degHu+ 1).(degHv + 1) (degHu+ 1) + (degHv + 1) ] + ∑ ef∈E(J [G]) [2 √ (degJ [G]e+ n2).(degJ [G]f + n2) (degJ [G]e+ n2) + (degJ [G]f + n2) ] + ∑ e∈V (J [G]) ∑ u∈V (H) [2 √ (degHu+ 1).(degJ(G)e+ n2) (degHu+ 1) + (degJ(G)e+ n2) ] = m1m2 [ 2 √ (degHu+ 1).(degHv + 1) (degHu+ 1) + (degHv + 1) ] +m1n2 [2 √ (degHu+ 1).(degJ [G]e+ n2) (degHu+ 1) + (degJ [G]e+ n2) ] + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]][2 √ (degJ [G]e+ n2).(degJ [G]f + n2) (degJ [G]e+ n2) + (degJ [G]f + n2) ] = m1m2 [ 2 √ (degHu+ 1).(degHv + 1) (degHu+ 1) + (degHv + 1) ] +m1n2 [ 2 √ (degHu+ 1).[(m1 − 1)− [degGu+ degGv − 2] + n2] (degHu+ 1) + [(m1 − 1)− [degGu+ degGv − 2] + n2] ] + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]] [ 2 √ [(m1 − 1)− [degGu+ degGv − 2] + n2].[(m1 − 1)− [degGu+ degGv − 2] + n2] [(m1 − 1)− [degGu+ degGv − 2] + n2] + [(m1 − 1)− [degGu+ degGv − 2] + n2] ] ≥ m1m2 [ 2 √ (∆H + 1).(∆H + 1) (∆H + 1) + (∆H + 1) ] + [ 2m1n2 √ (∆H + 1).[(m1 − 1)− [∆G + ∆G − 2] + n2] ∆H + 1 +m1 − 1− [∆G + ∆G − 2] + n2 ] M. Manjunath , V. Lokesha, Suvarna, S. Jain / Eur. J. Pure Appl. Math, 14 (2) (2021), 340-350 345 + [[ m1(m1 − 1) 2 ] −m1 [ ∆G + ∆G − 2 2 ]] [ 2 √ [(m1 − 1)− [∆G + ∆G − 2] + n2].[(m1 − 1)− [∆G + ∆G − 2] + n2] [(m1 − 1)− [∆G + ∆G − 2] + n2] + [(m1 − 1)− [∆G + ∆G − 2] + n2] ] . GA[℘] ≥ m1m2 + 2m1n2 √ (∆H + 1).[(m1 − 1)− 2[∆G − 1] + n2] ∆H +m1 − 2[∆G − 1] + n2 + m1(m1 − 1) 2 −m1(∆G − 1). Similarly, GA[℘] ≤ m1m2 + 2m1n2 √ (δH + 1).[(m1 − 1)− 2[δG − 1] + n2] δH +m1 − 2[δG − 1] + n2 + m1(m1 − 1) 2 −m1(δG − 1). Theorem 3. Let G and H are two simple connected graphs, then the bounds for the atom-bond connective index of ℘ are given by ABC[℘] ≥ m1m2 [√ 2∆H (∆H + 1)2 ] +m1n2 [√ ∆H +m1 − 2∆G + n2 [∆H + 1].[m1 − 1− 2[∆G − 1] + n2] ] + [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ][√ 2[m1 − 2∆G + n2] [(m1 − 1)− 2[∆G − 1] + n2]2 ] and ABC[℘] ≤ m1m2 [√ 2δH (δH + 1)2 ] +m1n2 [√ δH +m1 − 2δG + n2 [δH + 1].[m1 − 1− 2[δG − 1] + n2] ] + [[ m1(m1 − 1) 2 ] −m1(δG − 1) ][√ 2[m1 − 2δG + n2] [(m1 − 1)− 2[δG − 1] + n2]2 ] . Proof. Consider ABC[℘] = m1 ∑ uv∈E(H) [√ (degHu+ 1) + (degHv + 1)− 2 (degHu+ 1).(degHv + 1) ] + ∑ e∈V (J [G]) ∑ u∈V (H) [√ (degHu+ 1) + (degJ(G)e+ n2)− 2 (degHu+ 1).(degJ(G)e+ n2) ] + ∑ ef∈E(J [G]) [√ (degJ [G]e+ n2) + (degJ [G]f + n2)− 2 (degJ [G]e+ n2).(degJ [G]f + n2) ] = m1m2 [√ (degHu+ 1) + (degHv + 1)− 2 (degHu+ 1).(degHv + 1) ] +m1n2 [√ (degHu+ 1) + (degJ [G]e+ n2)− 2 (degHu+ 1).(degJ [G]e+ n2) ] M. Manjunath , V. Lokesha, Suvarna, S. Jain / Eur. J. Pure Appl. Math, 14 (2) (2021), 340-350 346 + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]][√ (degJ [G]e+ n2) + (degJ [G]f + n2)− 2 (degJ [G]e+ n2).(degJ [G]f + n2) ] = m1m2 [√ (degHu+ 1) + (degHv + 1)− 2 (degHu+ 1).(degHv + 1) ] + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]] [√ [(m1 − 1)− [degGu+ degGv − 2] + n2] + [(m1 − 1)− [degGu+ degGv − 2] + n2]− 2 [(m1 − 1)− [degGu+ degGv − 2] + n2].[(m1 − 1)− [degGu+ degGv − 2] + n2] ] +m1n2 [√ (degHu+ 1) + [(m1 − 1)− [degGu+ degGv − 2] + n2]− 2 (degHu+ 1).[(m1 − 1)− [degGu+ degGv − 2] + n2] ] ≥ m1m2 [√ (∆H + 1) + (∆H + 1)− 2 (∆H + 1).(∆H + 1) ] +m1n2 [√ ∆H + 1 +m1 − 1− [∆G + ∆G − 2] + n2 − 2 [∆H + 1].[m1 − 1− [∆G + ∆G − 2] + n2] ] + [[ m1(m1 − 1) 2 ] −m1 [ ∆G + ∆G − 2 2 ]] [√ [(m1 − 1)− [∆G + ∆G − 2] + n2] + [(m1 − 1)− [∆G + ∆G − 2] + n2]− 2 [(m1 − 1)− [∆G + ∆G − 2] + n2].[(m1 − 1)− [∆G + ∆G − 2] + n2] ] ≥ m1m2 [√ 2(∆H + 1)− 2 (∆H + 1)2 ] +m1n2 [√ ∆H +m1 − 2[∆G − 1] + n2 − 2 [∆H + 1].[m1 − 1− 2[∆G − 1] + n2] ] + [[ m1(m1 − 1) 2 ] −m1 [ 2(∆G − 1) 2 ]][√ 2[(m1 − 1)− 2[∆G − 1] + n2]− 2 [(m1 − 1)− 2[∆G − 1] + n2]2 ] ≥ m1m2 [√ 2(∆H + 1− 1) (∆H + 1)2 ] +m1n2 [√ ∆H +m1 − 2[∆G − 1 + 1] + n2 [∆H + 1].[m1 − 1− 2[∆G − 1] + n2] ] + [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ][√ 2[m1 − 1− 2[∆G − 1] + n2 − 1] [(m1 − 1)− 2[∆G − 1] + n2]2 ] . ABC[℘] ≥ m1m2 [√ 2∆H (∆H + 1)2 ] +m1n2 [√ ∆H +m1 − 2∆G + n2 [∆H + 1].[m1 − 1− 2[∆G − 1] + n2] ] + [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ][√ 2[m1 − 2∆G + n2] [(m1 − 1)− 2[∆G − 1] + n2]2 ] . Similarly, ABC[℘] ≤ m1m2 [√ 2δH (δH + 1)2 ] +m1n2 [√ δH +m1 − 2δG + n2 [δH + 1].[m1 − 1− 2[δG − 1] + n2] ] M. Manjunath , V. Lokesha, Suvarna, S. Jain / Eur. J. Pure Appl. Math, 14 (2) (2021), 340-350 347 + [[ m1(m1 − 1) 2 ] −m1(δG − 1) ][√ 2[m1 − 2δG + n2] [(m1 − 1)− 2[δG − 1] + n2]2 ] . Theorem 4. Let G and H are two simple connected graphs, then the bounds for the first zagreb index of ℘ are given by M1[℘] ≥ 2m1m2(∆H + 1) +m1n2[∆H +m1 − 2[∆G − 1] + n2] + 2 [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ] [m1 − 1− 2[∆G − 1] + n2] and M1[℘] ≤ 2m1m2(δH + 1) +m1n2[δH +m1 − 2[δG − 1] + n2] + 2 [[ m1(m1 − 1) 2 ] −m1(δG − 1) ] [m1 − 1− 2[δG − 1] + n2]. Proof. Consider M1[℘] = m1 ∑ uv∈E(H) [(degHu+ 1) + (degHv + 1)] + ∑ e∈V (J [G]) ∑ u∈V (H) [(degHu+ 1) + (degJ(G)e+ n2)] + ∑ ef∈E(J [G]) [(degJ [G]e+ n2) + (degJ [G]f + n2)] = m1m2[(degHu+ 1) + (degHv + 1)] +m1n2[(degHu+ 1) + (degJ [G]e+ n2)] + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]] [(degJ [G]e+ n2) + (degJ [G]f + n2)] = m1m2[(degHu+ 1) + (degHv + 1)] + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]] [[(m1 − 1)− [degGu+ degGv − 2] + n2] + [(m1 − 1)− [degGu+ degGv − 2] + n2]] +m1n2[(degHu+ 1) + [(m1 − 1)− [degGu+ degGv − 2] + n2]] ≥ m1m2[(∆H + 1) + (∆H + 1)] +m1n2[∆H + 1 +m1 − 1− [∆G + ∆G − 2] + n2] + [[ m1(m1 − 1) 2 ] −m1 [ ∆G + ∆G − 2 2 ]] [2[(m1 − 1)− [∆G + ∆G − 2] + n2]] ≥ 2m1m2(∆H + 1) +m1n2[∆H +m1 − 2[∆G − 1] + n2] + 2 [ m1(m1 − 1) 2 −m1 [ 2(∆G − 1) 2 ]] [m1 − 1− 2[∆G − 1] + n2]. M. Manjunath , V. Lokesha, Suvarna, S. Jain / Eur. J. Pure Appl. Math, 14 (2) (2021), 340-350 348 M1[℘] ≥ 2m1m2(∆H + 1) +m1n2[∆H +m1 − 2[∆G − 1] + n2] + 2 [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ] [m1 − 1− 2[∆G − 1] + n2]. Similarly, M1[℘] ≤ 2m1m2(δH + 1) +m1n2[δH +m1 − 2[δG − 1] + n2] + 2 [[ m1(m1 − 1) 2 ] −m1(δG − 1) ] [m1 − 1− 2[δG − 1] + n2]. Theorem 5. Let G and H are two simple connected graphs, then the bounds for the first reformulated zagreb index of ℘ are given by EM1[℘] ≥ 4m1m2∆ 2 H +m1n2[∆H +m1 − 2∆G + n2] 2 + 4 [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ] [m1 − 2∆G + n2] 2 and EM1[℘] ≤ 4m1m2δ 2 H +m1n2[δH +m1 − 2δG + n2] 2 + 4 [[ m1(m1 − 1) 2 ] −m1(δG − 1) ] [m1 − 2δG + n2] 2. Proof. Consider, EM1[℘] = m1 ∑ uv∈E(H) [(degHu+ 1) + (degHv + 1)− 2]2 + ∑ e∈V (J [G]) ∑ u∈V (H) [(degHu+ 1) + (degJ(G)e+ n2)− 2]2 + ∑ ef∈E(J [G]) [(degJ [G]e+ n2) + (degJ [G]f + n2)− 2]2 = m1m2[(degHu+ 1) + (degHv + 1)− 2]2 +m1n2[(degHu+ 1) + (degJ [G]e+ n2)− 2]2 + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]] [(degJ [G]e+ n2) + (degJ [G]f + n2)− 2]2 = m1m2[(degHu+ 1) + (degHv + 1)− 2]2 + [[ m1(m1 − 1) 2 ] −m1 [ degGu+ degGv − 2 2 ]] [[(m1 − 1)− [degGu+ degGv − 2] + n2] + [(m1 − 1)− [degGu+ degGv − 2] + n2]− 2]2 +m1n2[(degHu+ 1) + [(m1 − 1)− [degGu+ degGv − 2] + n2]− 2]2 ≥ m1m2[(∆H + 1) + (∆H + 1)− 2]2 + [[ m1(m1 − 1) 2 ] −m1 [ ∆G + ∆G − 2 2 ]] [[(m1 − 1)− [∆G + ∆G − 2] + n2] + [(m1 − 1)− [∆G + ∆G − 2] + n2]− 2]2 REFERENCES 349 +m1n2[∆H + 1 +m1 − 1− [∆G + ∆G − 2] + n2 − 2]2 ≥ m1m2[2(∆H + 1)− 2]2 + [[ m1(m1 − 1) 2 ] −m1 [ 2(∆G − 1) 2 ]] [2[(m1 − 1)− 2[∆G − 1] + n2]− 2]2 +m1n2[∆H +m1 − 2[∆G − 1] + n2 − 2]2 ≥ 4m1m2[∆H + 1− 1]2 + [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ] [2[m1 − 1− 2[∆G − 1] + n2 − 1]]2 +m1n2[∆H +m1 − 2[∆G − 1 + 1] + n2] 2 ≥ 4m1m2∆ 2 H +m1n2[∆H +m1 − 2∆G + n2] 2 + 4 [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ] [m1 − 2∆G + n2] 2. EM1[℘] ≥ 4m1m2∆ 2 H +m1n2[∆H +m1 − 2∆G + n2] 2 + 4 [[ m1(m1 − 1) 2 ] −m1(∆G − 1) ] [m1 − 2∆G + n2] 2. Similarly, EM1[℘] ≤ 4m1m2δ 2 H +m1n2[δH +m1 − 2δG + n2] 2 + 4 [[ m1(m1 − 1) 2 ] −m1(δG − 1) ] [m1 − 2δG + n2] 2. 3. Conclusion In this work, we considered ℘-graph and concentrated five important topological indices and determine their bounds. Similar way, researchers can considering different class of topological indices and determine their corresponding bounds for ℘-graph. Acknowledgements The authors thankful to Referee for improvement of the paper. References [1] E. Estrada. Atom-bond connectivity and the energetic of branched alkanes. Chemical Physics Letters, 463:422–425, 2008. [2] I. Gutman and N. Trinajstic. Graph theory and molecular obitals. Total π-electron energy of alternate hydrocarbons. Chemical Physics Letters, 17:535–538, 1972. REFERENCES 350 [3] Winer. H. Structural determination of paraffin boiling points. Journal of american chemical society, 69:17–20, 1947. [4] V. 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