Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 110 https://internationalpubls.com Extended Reverse R Degrees of Vertices and Extended Reverse R indices of Graphs 1T. Lavanya, 2K.A. Venkatesh, 3D. Amsaveni 1,3Department of Mathematics, 1Bharat Ratna Puratchi Thalaivar Dr. M.G.R. Government Arts and Science, Palacode-636808 2Myanmar Institute of Information Technology, Myanmar. 3Sri Sarada College for Women(Autonomous), Salem-636016. Email: 1tplaly@gmail.com, 2prof.kavenatesh@gmail.com and 3d_amsaveni@rediffmail.com Article History: Received: 18-05-2024 Revised: 14-06-2024 Accepted: 30-06-2024 Abstract A topological representation of a molecule is called molecular graph. A molecular graph is a collection of points representing the atoms in the molecule and set of lines represent the covalent bonds. Topological indices gather data from the graph of molecule and help to foresee properties of the concealing molecule. All the degree based topological indices have been defined through classical degree concept. In this paper, we define a novel degree concept for a vertex of a simple connected graph: Extended Reverse R degree and also, we define Extended Reverse R indices of a simple connected graph by using the Extended Reverse R degree concept. We compute the Extended Reverse R indices using the above contemporary degree concept for well-known simple connected graphs such as complete bipartite graph, Wheel graph, Generalized Peterson graph, Crown graph, Double star graph, and Windmill graph. Keywords: Reverse degree, Topological indices, extended reverse R degree, extended reverse R indices. AMS Mathematics Subject Classification (2020): 05C09, 05C07, 05C31,05C38. 1. Introduction A topological index is a mathematical invariant that characterize the chemical properties of a molecule. These indices are used in quantitative structure property relations (QSPR) research. Topological indices are important tools for analyzing some physicochemical properties of molecules without performing any experiment. The Wiener index W(G) is a distance-based topological invariant much used in the study of the structure-property and the structure-activity relationships of various classes of biochemically interesting compounds, which is introduced in 1947 for prognosticating boiling points by Harold Wiener [16]. As of now, myriad β€œMolecular descriptors” are being put forwarded. Recently, degree based topological indices are also formed a good correlation with chemical properties of a molecule. Some well-known degree based topological indices are Randic index, First and Second Zagreb indices, reformulated first and second Zagreb indices, Atom-Bond Connectivity index, Augmented Zagreb index, Harmonic index, Geometric-arithmetic index, Sum connectivity index are studied in [1-9], [11], [12], [15], [18] and [19]. The comparative testing of these well-known degree based topological indices were given in [10]. The concept of R degree of a vertex Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 111 https://internationalpubls.com and R index of a graph were introduced by Siileyman Ediz[17]. In this paper, the extended reverse β„› indices for well-known simple connected graphs such as Complete bipartite graph, Wheel graph, Generalized Peterson graph, Crown graph, Double star graph and Windmill graph are obtained. Throughout this paper only simple connected graphs were considered, that is connected graphs without self-loops and parallel edges. 2. EXTENDED REVERSE R INDICES The graph G = (V, E) = (V(G),E(G)) have the set of all vertices V(G) and the set of all edges E(G) respectively. The degree of the vertex v is defined as the number of edges incident with v and denoted by d(v). The set of all vertices which are adjacent to v is called the neighborhood of v and it is denoted by N(v). The reverse degree of a vertex v is RdV = π›₯ βˆ’ 𝑑(𝑣) + 1 where π›₯ is maximum degree of G. Definition:2.1 The reverse sum degree of v is defined as 𝑅𝑆𝑉 = βˆ‘ π‘…π‘‘π‘’π‘’βˆˆπ‘(𝑣) and the reverse multiplication degree of a vertex v is defined as 𝑅𝑀𝑣 = ∏ π‘…π‘‘π‘’π‘’βˆˆπ‘(𝑣) . Definition:2.2 The Extended Reverse β„› degree of a vertex v of a simple connected graph G is defined as πΈπ‘…π‘Ÿ(𝑣) = 𝑅𝑆𝑉 + 𝑅𝑀𝑣. Definition:2.3 Let 𝐺 = (𝑉, 𝐸) be a graph. (a) The Extended Reverse first index of a simple connected graph G defined as πΈπ‘…π‘Ÿ 1(𝐺) = βˆ‘ [πΈπ‘…π‘Ÿ(𝑣)]2 π‘£βˆˆπ‘‰(𝐺) . (b)The Extended Reverse Second index of a simple connected graph G defined as πΈπ‘…π‘Ÿ 2(𝐺) = βˆ‘ [πΈπ‘…π‘Ÿ(𝑒)πΈπ‘…π‘Ÿ(𝑣)]<𝑒𝑣>∈𝐸(𝐺) . (b) The Extended Reverse third index of a simple connected graph G defined as πΈπ‘…π‘Ÿ 3(𝐺) = βˆ‘ [πΈπ‘…π‘Ÿ(𝑒) + πΈπ‘…π‘Ÿ(𝑣)]<𝑒𝑣>∈𝐸(𝐺) . Hence, the extended reverse β„› indices are topological indices. 3. EXTENDED REVERSE β„› INDICES OF SOME GRAPHS In this section, the Complete bipartite graph, Wheel graph, Generalized Peterson graph, Crown graph, Double star graph and Windmill graph are characterized using the extended reverse β„› indices. Theorem 3.1 If πΎπ‘š,𝑛 is the Complete bipartite graph with n+m vertices, n > m and mn edges then π”Όβ„œβ„› 1 (πΎπ‘š,𝑛) = 𝑛[π‘š + 1]2 + π‘š[(𝑛 βˆ’ π‘š + 1)(𝑛 + (𝑛 βˆ’ π‘š + 1)π‘›βˆ’1)]2, π”Όβ„œβ„› 2 (πΎπ‘š,𝑛) = π‘šπ‘›{[π‘š + 1][(𝑛 βˆ’ π‘š + 1)(𝑛 + (𝑛 βˆ’ π‘š + 1)π‘›βˆ’1)]}, π”Όβ„œβ„› 3 (πΎπ‘š,𝑛) = π‘šπ‘›{[π‘š + 1] + [(𝑛 βˆ’ π‘š + 1)(𝑛 + (𝑛 βˆ’ π‘š + 1)π‘›βˆ’1)]}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 112 https://internationalpubls.com Proof: Let πΎπ‘š,𝑛 be a Complete bipartite graph (𝑉𝑖, 𝑉𝑗, 𝐸). The {𝑒1, 𝑒2, … , π‘’π‘š} βŠ‚ 𝑉𝑖 and {𝑣1, 𝑣2, … , 𝑣𝑛} βŠ‚ 𝑉𝑗 are any two sets of vertices. Here, (𝑒𝑖,𝑒𝑗), 𝑖 = 1,2,3, … , π‘š, 𝑗 = 1,2,3, … , 𝑛 are edges in E. A Complete bipartite graph with partitions of size |𝑉𝑖| = π‘š, |𝑉𝑗| = 𝑛, |𝑉(πΎπ‘š,𝑛)| = π‘š + 𝑛 and |𝐸(πΎπ‘š,𝑛)| = π‘šπ‘› Then the reverse vertex degree β„œπ‘‘π‘’π‘– = 1, β„œπ‘‘π‘£π‘— = 𝑛 βˆ’ π‘š + 1 , the reverse sum degree β„œπ‘†π‘’π‘– = 𝑛(𝑛 βˆ’ π‘š + 1), β„œπ‘†π‘£π‘— = π‘š and the reverse multiplication degree β„œπ‘€π‘’π‘– = (𝑛 βˆ’ π‘š + 1)𝑛, β„œπ‘€π‘£π‘— = 1. Then the extended reverse β„› degrees are π”Όβ„œβ„›π‘’π‘– 1 = (𝑛 βˆ’ π‘š + 1)[(𝑛 + (𝑛 βˆ’ π‘š + 1)π‘›βˆ’1)]2, π”Όβ„œβ„›π‘£π‘— 1 = π‘š + 1. Hence, the extended reverse β„› topological indices of πΎπ‘š,𝑛 are π”Όβ„œβ„› 1 (πΎπ‘š,𝑛) = 𝑛[π‘š + 1]2 + π‘š[(𝑛 βˆ’ π‘š + 1)(𝑛 + (𝑛 βˆ’ π‘š + 1)π‘›βˆ’1)]2, π”Όβ„œβ„› 2 (πΎπ‘š,𝑛) = π‘šπ‘›{[π‘š + 1][(𝑛 βˆ’ π‘š + 1)(𝑛 + (𝑛 βˆ’ π‘š + 1)π‘›βˆ’1)]}, π”Όβ„œβ„› 3 (πΎπ‘š,𝑛) = π‘šπ‘›{[π‘š + 1] + [(𝑛 βˆ’ π‘š + 1)(𝑛 + (𝑛 βˆ’ π‘š + 1)π‘›βˆ’1)]}. Theorem 3.2 If (π‘Šπ‘›) is the Wheel graph with n vertices n β‰₯ 4 then π”Όβ„œβ„› 1 (π‘Šπ‘›) = [(𝑛 βˆ’ 1)(𝑛 βˆ’ 3) + (𝑛 βˆ’ 3)π‘›βˆ’1]2 + (𝑛 βˆ’ 1)(𝑛 βˆ’ 2)4, π”Όβ„œβ„› 2 (π‘Šπ‘›) = (𝑛 βˆ’ 1)[{(𝑛 βˆ’ 1)(𝑛 βˆ’ 3) + (𝑛 βˆ’ 3)π‘›βˆ’1][𝑛 βˆ’ 2]2} + (𝑛 βˆ’ 2)4, π”Όβ„œβ„› 3 (π‘Šπ‘›) = (𝑛 βˆ’ 1)(𝑛 βˆ’ 3)π‘›βˆ’1 + 4𝑛3 βˆ’ 20𝑛2 + 31𝑛 βˆ’ 15. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 113 https://internationalpubls.com Proof: A Wheel graph Wn with n vertices and 2n-2 edges is obtained by connecting a single vertex to a vertices of a cycle of length n-1. The set of vertices {𝑣1, 𝑣2, 𝑣3, . . . , 𝑣𝑛} can be classified into two sets of vertices such that 𝑣1 and {𝑣𝑗 , 𝑗 = 1,2, … , 𝑛}. Then the reverse vertex degree β„œπ‘‘π‘£1 = 1, β„œπ‘‘π‘£π‘— = 𝑛 βˆ’ 3. The reverse sum degree β„œπ‘†π‘£1 = (𝑛 βˆ’ 1)(𝑛 βˆ’ 3), β„œπ‘†π‘£π‘— = 2𝑛 βˆ’ 5 and the reverse multiplication degree β„œπ‘€π‘£1 = (𝑛 βˆ’ 3)π‘›βˆ’1, β„œπ‘€π‘£π‘— = (𝑛 βˆ’ 3)2. Then the extended reverse β„› degrees are π”Όβ„œβ„›π‘£1 ⬚ = (𝑛 βˆ’ 1)(𝑛 βˆ’ 3) + (𝑛 βˆ’ 3)π‘›βˆ’1, π”Όβ„œβ„›π‘£π‘— ⬚ = (𝑛 βˆ’ 2)2. π”Όβ„œβ„› 1 (π‘Šπ‘›) = [(𝑛 βˆ’ 1)(𝑛 βˆ’ 3) + (𝑛 βˆ’ 3)π‘›βˆ’1]2 + (𝑛 βˆ’ 1)(𝑛 βˆ’ 2)4, π”Όβ„œβ„› 2 (π‘Šπ‘›) = (𝑛 βˆ’ 1)[{(𝑛 βˆ’ 1)(𝑛 βˆ’ 3) + (𝑛 βˆ’ 3)π‘›βˆ’1][𝑛 βˆ’ 2]2} + (𝑛 βˆ’ 2)4, π”Όβ„œβ„› 3 (π‘Šπ‘›) = (𝑛 βˆ’ 1)(𝑛 βˆ’ 3)π‘›βˆ’1 + 4𝑛3 βˆ’ 20𝑛2 + 31𝑛 βˆ’ 15. Theorem 3.3 If 𝐺𝑃𝑛,π‘˜ be the Generalized Peterson graph with n β‰₯ 3 and 1 ≀ π‘˜ ≀ ⌊ (π‘›βˆ’1) 2 βŒ‹ then π”Όβ„œβ„› 1 (𝐺𝑃𝑛,π‘˜) = 2𝑛(𝑛 + 1)2, π”Όβ„œβ„› 2 (𝐺𝑃𝑛,π‘˜) = 3𝑛(𝑛 + 1)2, π”Όβ„œβ„› 3 (𝐺𝑃𝑛,π‘˜) = 6𝑛(𝑛 + 1)2. Proof: The vertex and edge cardinality of Generalized Peterson graph is |𝑉(𝐺𝑃𝑛,π‘˜)| = 2𝑛, |𝐸(𝐺𝑃𝑛,π‘˜)| = 3𝑛 respectively. The reverse vertex degree degree β„œπ‘‘π‘£ = 1. The reverse sum degree β„œπ‘†π‘£ = 𝑛 and the reverse multiplication degree β„œπ‘€π‘£1 = 1. Then the extended reverse β„› degree π”Όβ„œπ‘£ ⬚ = (𝑛 + 1) . π”Όβ„œβ„› 1 (𝐺𝑃𝑛,π‘˜) = 2𝑛(𝑛 + 1)2, π”Όβ„œβ„› 2 (𝐺𝑃𝑛,π‘˜) = 3𝑛(𝑛 + 1)2, π”Όβ„œβ„› 3 (𝐺𝑃𝑛,π‘˜) = 6𝑛(𝑛 + 1)2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 114 https://internationalpubls.com Theorem 3.4 If 𝑆𝑛 is the Crown graph with 2n vertices and 𝑛(𝑛 βˆ’ 1) edges then π”Όβ„œβ„› 1 (𝑆𝑛) = 2𝑛3, π”Όβ„œβ„› 2 (𝑆𝑛) = 𝑛3(𝑛 βˆ’ 1), π”Όβ„œβ„› 3 (𝑆𝑛) = 2𝑛2(𝑛 βˆ’ 1). Proof: The Crown graph 𝑆𝑛 is a graph whose vertices can be subdivided into two sets of vertices {𝑒1, 𝑒2, 𝑒3, . . . , 𝑒𝑛} and {𝑣1, 𝑣2, 𝑣3, . . . , 𝑣𝑛} as 𝑣 and with an edge from 𝑒𝑖 to 𝑣𝑗 whenever 𝑖 β‰  𝑗. A size of Crown graph is |𝑉(𝑆𝑛)| = 2𝑛, |𝐸(𝑆𝑛)| = 𝑛(𝑛 βˆ’ 1). Then the reverse vertex degree degree β„œπ‘‘π‘£ = 1. The reverse sum degree β„œπ‘†π‘£ = 𝑛(𝑛 βˆ’ 1) and the reverse multiplication degree β„œπ‘€π‘£1 = 1. The extended reverse β„› degree π”Όβ„œπ‘…π‘£ ⬚ = 𝑛. Hence, the extended reverse β„› topological indices of 𝑆𝑛 are π”Όβ„œβ„› 1 (𝑆𝑛) = 2𝑛3, π”Όβ„œβ„› 2 (𝑆𝑛) = 𝑛3(𝑛 βˆ’ 1), π”Όβ„œβ„› 3 (𝑆𝑛) = 2𝑛2(𝑛 βˆ’ 1). Theorem 3.5 If Sm,n is the Double star graph with n+m+2 vertices (n > m) and n+m+1edges then π”Όβ„œβ„› 1 (π‘†π‘š,𝑛) = {(𝑛2 βˆ’ π‘š + 1)𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1} 2 {π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1) + 𝑛2 βˆ’ π‘š + 1}2 + 4(π‘š βˆ’ 1){𝑛 βˆ’ π‘š + 1}2 + 4𝑛 βˆ’ 4, π”Όβ„œβ„› 2 (π‘†π‘š,𝑛) = 2(π‘š βˆ’ 1){(𝑛(π‘šβˆ’1) + (𝑛(π‘š βˆ’ 1) + 1))(𝑛 βˆ’ π‘š + 1)} + 2(𝑛 βˆ’ 1) (𝑛2 βˆ’ π‘š + 1 + π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1)) + (𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1)(π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1) + 𝑛2 βˆ’ π‘š + 1), π”Όβ„œβ„› 3 (π‘†π‘š,𝑛) = (π‘š βˆ’ 1){𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1 + 2(𝑛 βˆ’ π‘š + 1)} + (𝑛 βˆ’ 1){(𝑛2 βˆ’ π‘š + 3 + π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1)) + 2} + (𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1) + (π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1) + 𝑛2 βˆ’ π‘š + 1). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 115 https://internationalpubls.com Proof: The Double star graph Sm,n n, m β‰₯ 2 and (n > m). Here, |𝑉(π‘†π‘š,𝑛)| = 𝑛 + π‘š + 2, |𝐸(π‘†π‘š,𝑛)| = 𝑛 βˆ’ π‘š + 1. Then, reverse vertex degrees are β„œπ‘‘π‘‰πΌ = 𝑛 βˆ’ π‘š + 1, β„œπ‘‘π‘‰πΌπΌ = 𝑛 and β„œπ‘‘π‘‰πΌπΌπΌ = 1 where 𝑉𝐼 is the central vertex of m star, is the pendent vertex of m star, is the central vertex of n star and is the pendent vertex of n star graph. The reverse sum degrees are β„œπ‘†π‘‰πΌ = π‘šπ‘› βˆ’ 𝑛 + 1, β„œπ‘†π‘‰πΌπΌ = 𝑛 βˆ’ π‘š + 1, β„œπ‘†π‘‰πΌπΌπΌ = 𝑛2 βˆ’ π‘š + 1, β„œπ‘†π‘‰πΌπ‘‰ = 1 and the reverse multiplication degree β„œπ‘€π‘‰πΌ = (𝑛2 βˆ’ π‘š + 1)π‘›π‘šβˆ’1, β„œπ‘€π‘‰πΌπΌ = 𝑛 βˆ’ π‘š + 1, β„œπ‘€π‘‰πΌπΌπΌ = (𝑛 βˆ’ π‘š + 1)π‘›π‘›βˆ’1, β„œπ‘€π‘‰πΌπ‘‰ = 1. Then the extended reverse β„› degrees are π”Όβ„œβ„›π‘£πΌ ⬚ = (𝑛2 βˆ’ π‘š + 1)𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1, π”Όβ„œβ„›π‘‰πΌπΌ ⬚ = 2(𝑛 βˆ’ π‘š + 1), π”Όβ„œβ„›π‘‰πΌπΌπΌ ⬚ = π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1) + 𝑛2 βˆ’ π‘š + 1, π”Όβ„œβ„›π‘‰πΌπ‘‰ ⬚ = 2. Hence, the extended reverse β„› topological indices of π‘†π‘š,𝑛 are π”Όβ„œβ„› 1 (π‘†π‘š,𝑛) = {(𝑛2 βˆ’ π‘š + 1)𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1} 2 {π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1) + 𝑛2 βˆ’ π‘š + 1}2 + 4(π‘š βˆ’ 1){𝑛 βˆ’ π‘š + 1}2 + 4𝑛 βˆ’ 4, π”Όβ„œβ„› 2 (π‘†π‘š,𝑛) = 2(π‘š βˆ’ 1){(𝑛(π‘šβˆ’1) + (𝑛(π‘š βˆ’ 1) + 1))(𝑛 βˆ’ π‘š + 1)} + 2(𝑛 βˆ’ 1)(𝑛2 βˆ’ π‘š + 1 + π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1)) + (𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1)(π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1) + 𝑛2 βˆ’ π‘š + 1), π”Όβ„œβ„› 3 (π‘†π‘š,𝑛) = (π‘š βˆ’ 1){𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1 + 2(𝑛 βˆ’ π‘š + 1)} + (𝑛 βˆ’ 1){(𝑛2 βˆ’ π‘š + 3 + π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1)) + 2} + (𝑛(π‘šβˆ’1) + 𝑛(π‘š βˆ’ 1) + 1) + (π‘›π‘›βˆ’1(𝑛 βˆ’ π‘š + 1) + 𝑛2 βˆ’ π‘š + 1). Theorem 3.6 If π‘Šπ‘š (𝑛) is the Windmill graph with m, n β‰₯ 2 then π”Όβ„œβ„› 1 (π‘Šπ‘š (𝑛)) = {𝑛(π‘š βˆ’ 1)[(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1] + [(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1]𝑛(π‘šβˆ’1) } 2 + 𝑛(π‘š βˆ’ 1)[2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3]2, π”Όβ„œβ„› 2 (π‘Šπ‘š (𝑛)) = 𝑛(π‘š βˆ’ 1){(𝑛(π‘š βˆ’ 1)[(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1] + [(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1]𝑛(π‘šβˆ’1)) (2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3)} + [2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3]2, π”Όβ„œβ„› 3 (π‘Šπ‘š (𝑛)) = 𝑛(π‘š βˆ’ 1){(𝑛(π‘š βˆ’ 1)[(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1] + [(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1]𝑛(π‘šβˆ’1)) + (2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3)} + [2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3]2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 116 https://internationalpubls.com Proof: The Windmill graph π‘Šπ‘š (𝑛) is an undirected graph constructed for π‘š, 𝑛 β‰₯ 2 by joining n copies of the complete graph m at a shared universal vertex and |𝑉(π‘Šπ‘š (𝑛) )| = (π‘š βˆ’ 1)𝑛 + 1, |𝐸(π‘Šπ‘š (𝑛) )| = π‘šπ‘›(π‘šβˆ’1) 2 . The reverse vertex degrees are β„œπ‘‘π‘‰πΌ = 1, β„œπ‘‘π‘‰πΌπΌ = (π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1 where 𝑉𝐼 is the central vertex, 𝑉𝐼𝐼 is the vertices which are all adjacent to the central vertex. The reverse sum degrees are β„œπ‘†π‘‰πΌ = 𝑛(π‘š βˆ’ 1)[(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1], β„œπ‘†π‘‰πΌπΌ = (π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 2 and the reverse multiplicative degrees are β„œπ‘€π‘‰πΌ = [(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1]𝑛(π‘šβˆ’1), β„œπ‘€π‘‰πΌπΌ = (π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1. Then the extended reverse β„› degrees are π”Όβ„œβ„›π‘£πΌ ⬚ = 𝑛(π‘š βˆ’ 1)[(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1] + [(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1]𝑛(π‘šβˆ’1), π”Όβ„œβ„›π‘‰πΌπΌ ⬚ = 2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3. Hence, the extended reverse β„› topological indices of π‘Šπ‘š (𝑛) are π”Όβ„œβ„› 1 (π‘Šπ‘š (𝑛)) = {𝑛(π‘š βˆ’ 1)[(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1] + [(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1]𝑛(π‘šβˆ’1) } 2 + 𝑛(π‘š βˆ’ 1)[2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3]2, π”Όβ„œβ„› 2 (π‘Šπ‘š (𝑛)) = 𝑛(π‘š βˆ’ 1){(𝑛(π‘š βˆ’ 1)[(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1] + [(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1]𝑛(π‘šβˆ’1)) (2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3)} + [2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3]2, π”Όβ„œβ„› 3 (π‘Šπ‘š (𝑛)) = 𝑛(π‘š βˆ’ 1){(𝑛(π‘š βˆ’ 1)[(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1] + [(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 1]𝑛(π‘šβˆ’1)) + (2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3)} + [2(π‘š βˆ’ 1)(𝑛 βˆ’ 1) + 3]2. 4. Conclusion In this paper, the brand new degree based topological indices such as extended reverse R indices are elucidated using the reverse sum degree, reverse multiplicative degree and extended reverse R degree. Make use of, extended reverse R indices for the Complete bipartite graph, Wheel graph, Generalized Peterson graph, Crown graph, Double star graph and Windmill graph are structurally Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 117 https://internationalpubls.com characterized. This study may be regarded as an introduction to the topic and may seek to identify further recourse for simple connected graphs. Also one could concentrate on Chemical graphs. Acknowledgement. This research work funded by β€˜University Grand Commission(UGC)’ India under the Rajiv Gandhi National Fellowship (RGNF) Scheme and the scheme no: F117.1/201617/RGNF201517SCTAM27633/(SAIII/Website). REFERENCES [1] Das, K.C.; Xu K.; Nam, J.: On Zagreb indices of graphs, Front. Math. China, 10 562-582 (2015). 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