EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6212 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bounds of Geodetic-Wiener Index on Spirocyclic Graphs Rosalio G. Artes Jr.1,2, R. U. Gobithaasan1,∗, Roslan Hasni3, Nader Jafari Rad4 1 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia 2 Department of Mathematics, College of Arts and Sciences, Mindanao State University - Tawi-Tawi College of Technology and Oceanography, 7500 Bongao, Tawi-Tawi, Philippines 3 Special Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Com- puter Science and Mathematics, Universiti Malaysia Terengganu, 21030 UMT Kuala Nerus, Terengganu, Malaysia 4 Department of Mathematics, Shahed University, Tehran, Iran Abstract. Wiener index has been extensively studied for several decades because of its applica- tions in chemistry. Many variants of Wiener index were defined and their corresponding bounds were explored. In this work, we introduced the concept of geodetic-Wiener index by considering the number of geodesics between any pair of vertices. We used the concept of projection of a ver- tex to a subgraph to decompose the structure into subtrees. Simple spirocyclic graphs are bicyclic graphs whose cycles share a common vertex. Using the idea of partial Wiener index, we determined the bounds of geodetic-Wiener index with respect to other distance-based topological indices for simple spirocyclic graphs. 2020 Mathematics Subject Classifications: 05C30, 05C92 Key Words and Phrases: Topological index, geodetic-Wiener index, spirocyclic graphs 1. Introduction Topological indices (TIs) are 2D descriptors that consider the internal atomic struc- ture of compounds [1]. They incorporate data on molecular size, shape, branching, oc- currence of heteroatoms, and multiple bonds into numeric values  [1]. In 1947, Wiener [2] calculated boiling point of paraffin by using a distance-based TI called the Wiener index, which is the sum of distances between all vertex pairs in a graph. Since benzenoid hydrocarbons are fascinating the huge significance of theoretical chemists, the theory of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6212 Email addresses: rosalioartes@msutawi-tawi.edu.ph (R. G. Artes Jr.), gobithaasan@usm.my (R. U. Gobithaasan), hroslan@umt.edu.my (R. Hasni), n.jafarirad@shahed.ac.ir (N. Jafari Rad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 2 of 20 the Wiener index of the respective molecular graphs have been extensively developed in the last three decades [2]. Although the Wiener index is the oldest topological index, later on some distance-based topological indices have been defined such as hyper-Wiener index [3], Gutman index [4], Schultz index [5–9], Harary index [10, 11], additively weighted Harary index [12] and multiplicatively weighted Harary index [13]. The idea of Wiener index was introduced by Harold Wiener in 1947 [2]. The Wiener index of a connected graph G as defined in [2] is given by W (G) = ∑ {u,v}⊆V (G) dG(u, v). Recently, Lin [14] made a survey of all the extremal results on the Wiener index of trees from 2014 and presented some open problems. The generalized Wiener index of a a connected graph G was introduced by Martinez- Perez and Rodriguez [15] and is given by Wλ(G) = ∑ {u,v}⊆V (G) d(u, v)λ, where λ is a real number. The study of hyper-Wiener index of a connected graph G was pioneered by Alhevaz et al. [16] and is given by WW (G) = ∑ {u,v}⊆V (G) ( d(u, v) + 1 2 ) = 1 2 ∑ {u,v}⊆V (G) [d(u, v)2 + d(u, v)]. The Gutman index of G, denoted by Gut(G), which was introduced by Ivan Gutman in 1994 [17], is given by Gut(G) = ∑ {u,v}⊆V (G) [dudv]d(u, v). In 2014, Mazorodze et al. [18] established asymptotically sharp bound of the Gutman index for graphs without pendant vertices. The Schultz index of G [19] and is given by S(G) = ∑ {u,v}⊆V (G) [du + dv]d(u, v). The Harary index of G [11] is given by H(G) = ∑ {u,v}⊆V (G) 1 d(u, v) . R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 3 of 20 The additively weighted Harary index of G [12] is given by HA(G) = ∑ {u,v}⊆V (G) du + dv d(u, v) . The multiplicatively weighted Harary index of G was also proposed by Alizadeh et al. [12] as a modification of the Harary index and is given by HM (G) = ∑ {u,v}⊆V (G) dudv d(u, v) . Let U and U? be nonempty subsets of V (G). The partial Wiener Index of G [20] is given by W (U,U?;G) = ∑ (u,v)∈U×U? d(u, v). In 2017, Jamil [21] first derive closed-form formulas for some distance based topological indices for double graphs in terms of the original graph. Moreover, these formulas are applied for several special kinds of graphs, such as, the complete graph, the path and the cycle. In 2019, Dobrynin [22] esablished the Wiener index of uniform hypergraphs induced by trees. In the following year, Dobrynin and Estaji [23] investigated the Wiener index of hexagonal chains under some operations on the corresponding binary vectors. The obtained results may be useful in studying of topological indices for sets of hexagonal chains induced by algebraic constructions. 2. Spirocyclic Compounds and Applications Spiro compounds are organic compounds that have two or more rings with one com- mon atom, the spiro atom. The spiro atom gives a rigid geometry to the rings, which can impose shape and properties on the molecule. Spirocyclic compunds are an interest- ing group of organic compounds with distinct structural properties, making them suitable for application in many fields, especially medicinal chemistry. Spirocylic graphs represent a specific class of graphs characterized by unique structural features. These graphs, particularly in the theory of spiro compounds in organic chem- istry, are characterized by two or more cycles connected in one common vertex. This particular connectivity pattern imbues the molecules that correspond to it with distinc- tive properties, thus making them applicable to various chemical and potentially other scientific purposes. If a graph contains exactly two cycles, then we call it simple spirocyclic graph [24]. In drug discovery, an essential property of spirocycles is their natural capacity to project functionality to the third dimension [25]. This three-dimensional nature is essen- tial for drug-target interactions since drugs need to fit physically into biological targets, e.g., proteins, which occur in a much more specific and efficient manner than planar sys- R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 4 of 20 Figure 1: Skeletal formula and molecular structure of spirobicyclohexane tems [25]. An example of a spiro compound with a simple spirocyclic structure is the spirobicyclohexane (C11H20), with skeletal formula and molecular structure [26] shown in Figure 1. Spirobicyclohexane is a new organic compound whose structure is described by a bi- cyclic arrangement of two six-membered carbon rings in a spiro orientation. The impli- cation is that the two rings share a common axis, each with three carbon atoms that are not directly bonded. Spirobicyclohexane has a symmetrical and rigid structure, which can be one factor that influences its physical and chemical properties. It is often used as a build- ing block to construct more complex organic molecules. It is used in numerous ap- plications, including pharmaceuticals and materials science because it can impart spe- cific stereochemistry and stability to the molecules it is incorporated into [27]. Bicyclic graphs are graphs that contain exactly two cycles. Simple spirocyclic graphs are bicyclic graphs whose cycles share a common vertex called the spiro vertex. 3. Geodetic-Wiener Index Formulation The study of geodesics is a graph is an important graph-theoretic property to be con- sidered in topological properties of a graph [28]. For example, in a real-world scenario, the idea of considering different shortest routes from station A to station B can be mod- eled by the number of geodesics between these two stations. In this particular scenario, transporting goods from A to B could take less time by utilizing different transportation services on different shortest routes. The volume of goods that can be transported at the same time is determined by the number of shortest routes from the initial station to the terminal station. In general, the transport of energy from one point to another can be represented by geodesics in a network. These scenarios give us some motivations to study the number of geodesics in a graph and integrate the concept with the Wiener index. The distance d(u, v) between two vertices u and v in a connected graph G is the length of a u-v geodesic in G. A u-v geodesic in G is a shortest path joining u and v in G [29]. R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 5 of 20 The degree of a vertex u in G is denoted by du = dG(u) [30]. The minimum degree δ and the maximum degree ∆ are, respectively, defined as follows [31]: δ = min v∈V (G) { dG(v)}, ∆ = max v∈V (G) { dG(v)}. The following formulation can be found in [32]. We now introduce a Wiener-type topological index. Consider the function α : V (G)× V (G) −→ N, where α(u, v) is counts the number of geodesics between vertices u and v in a graph G as defined in [29]. The geodetic-Wiener index (g-Wiener index) of G, as defined in [32], is given by Wg(G) = ∑ {u,v}⊆V (G) α(u, v)d(u, v). In [32], we establish several bounds for geodetic-Wiener index for unicyclic graphs in terms of other distance-based topological indices. 4. The Partitioning Definition 1. Let G be a connected graph. The projection of u ∈ V (G) onto a subgraph H of G as the set PH(u) = {w ∈ V (H) : d(u,w) = d(u, V (H))} = {w ∈ V (H) : d(u,w) ≤ d(u, z) for any z ∈ V (H)} If PH(u) = {w}, we simply write PH(u) = w. Consider a simple spirocyclic graph G with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2. For each i ∈ {1, 2, . . . , 2t1}, let Ui = {u ∈ V (G) : PC2t1∪C2t2 (u) = ui}. Similarly, for each j ∈ {1, 2, . . . , 2t2}, let Vj = {v ∈ V (G) : PC2t1∪C2t2 (v) = vj}. Then the projection operator onto the union of C2t1 and C2t2 generates the tree-partition {U1 = V1, U2, . . . , U2t1 , V2, V3, . . . , V2t2} of V (G). This means that 〈Ui〉 and 〈Vj〉 are trees in G for each i ∈ {1, 2, . . . , 2t1} and j ∈ {1, 2, . . . , 2t2}. We call the set {U1 = V1, U2, . . . , U2t1 , V2, V3, . . . , V2t2} the tree-partition of V (G) with respect to its projection onto the union of cycles C2t1 and C2t2 . In the above illustration, PC6∪C4(a) = PC6∪C4(b) = PC6∪C4(c) = PC6∪C4(d) = PC6∪C4(u4) = u4. This gives U4 = {a, b, c, d, u4}. Also, U1 = {u1} and U3 = {e, u3}. Moveover, the pro- jection of f onto C6 ∪ C4 is v2. Note also that α(u4, v3) = 4. R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 6 of 20 .......................................................................... ..... ... . .. .. ... . .. .. .. . . .. . . ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ....... ..... ... . .. .. ... . .. .. .. . . .. . . .................................................................................................. ....... ..... ... . .. .. ... . .. .. .. . . .. . . .......................................................................... ..... ... . .. .. ... . .. .. .. . . .. . . ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ....... ..... ... . .. .. ... . .. .. .. . . .. . . .................................................................................................. ....... ..... ... . .. .. ... . .. .. .. . . .. . . ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ....... ..... ... . .. .. ... . .. .. .. . . .. . . ....... ..... ... . .. .. ... . .. .. .. . . .. . . .......................................................................... ..... ... . .. .. ... . .. .. .. . . .. . . .................................................................................................. ....... ..... ... . .. .. ... . .. .. .. . . .. . . ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ....... ..... ... . .. .. ... . .. .. .. . . .. . . .......................................................................... ..... ... . .. .. ... . .. .. .. . . .. . . .................................................................................................. ....... ..... ... . .. .. ... . .. .. .. . . .. . . ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ....... ..... ... . .. .. ... . .. .. .. . . .. . . .................................................................................................. ....... ..... ... . .. .. ... . .. .. .. . . .. . . ....... ..... ... . .. .. ... . .. .. .. . . .. . . .......................................................................... ..... ... . .. .. ... . .. .. .. . . .. . . .......................................................................... ..... ... . .. .. ... . .. .. .. . . .. . . ....... ..... ... . .. .. ... . .. .. .. . . .. . . ................................................................... ....... ..... ... . .. .. ... . .. .. .. . . .. . . ....... ..... ... . .. .. ... . .. .. .. . . .. . . a C6 C4u4 u1 v2 c b f d e u3 v3 Figure 2: Spirocyclic graph with C6 and C4 The geodetic-Wiener index and Wiener index differ only on the number of geodesics between pairs of vertices. Hence, we will consider those pairs where the number of geodesics is at least two. For u, v ∈ V (G), the value of α(u, v) is determined by the following classes of subsets of V (G). Lemma 1. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2, and {U1 = V1, U2, . . . , U2t1 , V2, V3, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of cycles C2t1 and C2t2. Then (i) α(u, v) = 2 if u ∈ Ui and v ∈ Ui+t1, for i ∈ {1, 2, . . . , t1}. (ii) α(u, v) = 2 if u ∈ Vj and v ∈ Vj+t2, for j ∈ {1, 2, . . . , t2}. (iii) α(u, v) = 2 if u ∈ Vt2+1 and v ∈ Ui, for i ∈ {2, . . . , t1}. (iv) α(u, v) = 2 if u ∈ Ut1+1 and v ∈ Vj, for j ∈ {2, . . . , t2}. (v) α(u, v) = 4 if u ∈ Ut1+1 and v ∈ Vt2+1. Moreover, if (u, v) is not in the above categories, then α(u, v) = 1. Proof. Let C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ] be the cycles in G, for some natural numbers t1, t2 ≥ 2, and {U1 = V1, U2, . . . , U2t1 , V2, V3, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of cycles C2t1 and C2t2 . (i) Fix i ∈ {1, 2, . . . , t1}. Let u ∈ Ui and v ∈ Ui+t1 . α(u, v) = 2 if u ∈ Ui and v ∈ Ui+t1 . Then there exist ui, ui+t1 in C2t1 such that PC2t1∪C2t2 (u) = ui and PC2t1∪C2t2 (v) = ui+t1 . Note that there are exactly 2 geodesics from ui to ui+t1 . Consequently, there are exactly 2 geodesics from u to v. Accordingly, α(u, v) = 2. (ii) The proof is similar to (i). (iii) Note that any geodesic from C2t2 to C2t1 passes through V1 = U1. Here, there are exactly 2 geodesics from a vertex in Vt2+1 to any vertex in V1 = U1. Moreover, for each i ∈ {2, . . . , t1}, there is only one shortest path from a vertex in U1 to a vertex in Ui. Consequently, if u ∈ Vt2+1 and v ∈ Ui, for i ∈ {2, . . . , t1}, then α(u, v) = 2. R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 7 of 20 (iv) Similar arguments with (iii) gives the desired result. (v) Let u ∈ Ut1+1 and v ∈ Vt2+1. Then any geodesic from u to v passes through u1 = v1. From (i), there are exactly 2 geodesics from u to u1. Similarly, from (ii), there are exactly 2 geodesics from v to v1. Accordingly, α(u, v) = 4. The following gives the exact relation between g-Wiener and Wiener indices. Lemma 2. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2. Then Wg(G) = W (G) + t1∑ i=1 W (Ui, Ui+t1 ;G) + t2∑ j=1 W (Vj , Vj+t2 ;G) + 2t1∑ i=2 W (Vt2+1, Ui;G) + 2t2∑ j=2 W (Ut1+1, Vj ;G) +W (Ut1+1, Vt2+1;G). where {U1 = V1, U2, . . . , U2t1 , V2, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of the cycles C2t1 and C2t2. Proof. The geodesics in Lemma 1(i) are covered in the first and the second quantities. Similarly, the geodesics in Lemma 1(ii) are covered in the first and the third quantities. Lemma 1(iii) gives the fourth and first quantities. Similarly, Lemma 1(iv) are covered in the fifth and the first quantities. The 4 geodesics in Lemma 1(v) are covered in the sixth, first, fourth, and fifth quantities. The result follows. 5. Bounds of g-Wiener Index with Wiener Index The following result establishes relation between g-Wiener index and Wiener index as defined in [2] for simple spirocyclic graphs. Theorem 1. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2. Then W (G) +At1 +Bt2 ≤ Wg(G) ≤ W (G) + Cdiam(G), where {U1 = V1, U2, . . . , U2t1 , V2, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of the cycles C2t1 and C2t2, and A = t1∑ i=1 |Ui||Ui+t1 |+ 2t2∑ j=2 |Ut1+1||Vj |+ |Ut1+1||Vt2+1|, R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 8 of 20 B = t2∑ j=1 |Vj ||Vj+t2 |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|, and C = t1∑ i=1 |Ui||Ui+t1 |+ t2∑ j=1 |Vj ||Vj+t2 |+ 2t2∑ j=2 |Ut1+1||Vj |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. Proof. Let t1, t2 ≥ 2 be a natural numbers. Consider the even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [v1, v2, . . . , v2t2 ] in G. Note that {u, v} ⊆ V (G), d(u, v) ≤ diam(G). Using Lemma 2, Wg(G) = ∑ {u,v}⊆V (G) α(u, v)d(u, v) = W (G) + t1∑ i=1 W (Ui, Ui+t1 ;G) + t2∑ j=1 W (Vj , Vj+t2 ;G) + 2t1∑ i=2 W (Vt2+1, Ui;G) + 2t2∑ j=2 W (Ut1+1, Vj ;G) +W (Ut1+1, Vt2+1;G) = W (G) + t1∑ i=1 ∑ (u,v)∈Ui×Ui+t1 d(u, v) + t2∑ j=1 ∑ (u,v)∈Vj×Vj+t2 d(u, v) + 2t1∑ i=2 ∑ (u,v)∈Vt2+1×Ui d(u, v) + 2t2∑ j=2 ∑ (u,v)∈Ut1+1×Vj d(u, v) + ∑ (u,v)∈Ut1+1×Vt2+1 d(u, v) ≤ W (G) + diam(G) t1∑ i=1 |Ui||Ui+t1 |+ diam(G) t2∑ j=1 |Vj ||Vj+t2 | +diam(G) 2t1∑ i=2 |Vt2+1||Ui|+ diam(G) 2t2∑ j=2 |Ut1+1||Vj |+ diam(G)|Ut1+1||Vt2+1| = W (G) + Cdiam(G) where C = t1∑ i=1 |Ui||Ui+t1 |+ t2∑ j=1 |Vj ||Vj+t2 |+ 2t2∑ j=2 |Ut1+1||Vj |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. Now, for (u, v) ∈ Ui × Ui+t1 , d(u, v) ≥ t1. Also, for (u, v) ∈ Vj × Vj+t2 , d(u, v) ≥ t2 . Moreover, if (u, v) ∈ Vt2+1 × Ui for i ∈ {2, 3, . . . , 2t1}, d(u, v) ≥ t2 since a u − v geodesic passes through V1 and the distance between V1 and Vt2+1 is t2. Similarly, if R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 9 of 20 (u, v) ∈ Ut1+1×Vj for j ∈ {2, 3, . . . , 2t2}, d(u, v) ≥ t1 since a u−v geodesic passes through U1 and the distance between U1 and Ut1+1 is t1. Moreover, for (u, v) ∈ Ut1+1 × Vt2+1, d(u, v) ≥ t1 + t2. Applying Lemma 2 gives Wg(G) = ∑ {u,v}⊆V (G) α(u, v)d(u, v) = W (G) + t1∑ i=1 W (Ui, Ui+t1 ;G) + t2∑ j=1 W (Vj , Vj+t2 ;G) + 2t1∑ i=2 W (Vt2+1, Ui;G) + 2t2∑ j=2 W (Ut1+1, Vj ;G) +W (Ut1+1, Vt2+1;G) = W (G) + t1∑ i=1 ∑ (u,v)∈Ui×Ui+t1 d(u, v) + t2∑ j=1 ∑ (u,v)∈Vj×Vj+t2 d(u, v) + 2t1∑ i=2 ∑ (u,v)∈Vt2+1×Ui d(u, v) + 2t2∑ j=2 ∑ (u,v)∈Ut1+1×Vj d(u, v) + ∑ (u,v)∈Ut1+1×Vt2+1 d(u, v) ≥ W (G) + t1 t1∑ i=1 |Ui||Ui+t1 |+ t2 t2∑ j=1 |Vj ||Vj+t2 | +t2 2t1∑ i=2 |Vt2+1||Ui|+ t1 2t2∑ j=2 |Ut1+1||Vj |+ (t1 + t2)|Ut1+1||Vt2+1| = W (G) +At1 +Bt2, where A = t1∑ i=1 |Ui||Ui+t1 |+ 2t2∑ j=2 |Ut1+1||Vj |+ |Ut1+1||Vt2+1| and B = t2∑ j=1 |Vj ||Vj+t2 |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. The proof is complete. Theorem 2. Let G be a simple spirocyclic graph with minimum degree δ = 2 and with even cycles C2t1 and C2t2, for some natural numbers t1, t2 ≥ 2. Then Wg(G) = W (G) + 2(t1 + t2) 2. Proof. Let C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [v1, v2, . . . , v2t2 ] be the cycles in G. Since δ = 2, G has no pendant vertices. Consequently, G is a vertex-gluing of C2t1 R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 10 of 20 and C2t2 . Assume that u1 = v1, the spiro vertex. Consider the tree-partition {V1 = U1, U2, . . . , U2t1 , V2, V3, . . . , V2t2} of V (G) using its projection onto C2t1 ∪ C2t2 . That is, Ui = {u ∈ V (G) : PC2t1∪C2t2 (u) = ui} for i ∈ {1, 2, . . . , 2t1} and Vj = {v ∈ V (G) : PC2t1∪C2t2 (v) = vj} for j ∈ {1, 2, . . . , 2t2}. Then U1 = V1 = {u1 = v1}, U2 = {u2}, U3 = {u3}, . . . , U2t1 = {u2t1}, V2 = {v2}, V3 = {v3}, . . . , V2t2 = {v2t2}. Hence, for each i ∈ {1, 2, . . . , 2t1}, |Ui| = 1. Similarly, for each j ∈ {1, 2, . . . , 2t2}, |Vj | = 1. For each i ∈ {2, 3, . . . , t1}, d(U1, Ui) = d(U1, U2t1−i+2) = i− 1. Hence, 2t1∑ i=2 ∑ (u,v)∈Vt2+1×Ui d(u, v) = (t2 + t1) + 2 t1∑ i=2 (t2 + i− 1) = 2t1t2 + t21 − t2. Similarly, for each j ∈ {2, 3, . . . , t2}, d(V1, Vj) = d(V1, V2t2−j+2) = j − 1. Thus, 2t2∑ j=2 ∑ (u,v)∈Ut1+1×Vj d(u, v) = (t1 + t2) + 2 t2∑ j=2 (t1 + j − 1) = 2t2t1 + t22 − t1. By Lemma 2, we have Wg(G) = ∑ {u,v}⊆V (G) α(u, v)d(u, v) = W (G) + t1∑ i=1 W (Ui, Ui+t1 ;G) + t2∑ j=1 W (Vj , Vj+t2 ;G) + 2t1∑ i=2 W (Vt2+1, Ui;G) + 2t2∑ j=2 W (Ut1+1, Vj ;G) +W (Ut1+1, Vt2+1;G) = W (G) + t1∑ i=1 ∑ (u,v)∈Ui×Ui+t1 d(u, v) + t2∑ j=1 ∑ (u,v)∈Vj×Vj+t2 d(u, v) + 2t1∑ i=2 ∑ (u,v)∈Vt2+1×Ui d(u, v) + 2t2∑ j=2 ∑ (u,v)∈Ut1+1×Vj d(u, v) + ∑ (u,v)∈Ut1+1×Vt2+1 d(u, v) = W (G) + t21 + t22 + [2t1t2 + t21 − t2] + [2t2t1 + t22 − t1] + (t1 + t2) = W (G) + 2(t1 + t2) 2. R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 11 of 20 6. Bounds of g-Wiener Index with Gutman Index The following result establishes the extremal bounds for the geodetic-Wiener index of a simple spirocyclic graph G containing even cycles in terms of the Gutman index of G as defined in [17]. Theorem 3. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2. Then 1 ∆2 Gut(G) +At1 +Bt2 < Wg(G) < 1 δ2 Gut(G) + Cdiam(G), where {U1 = V1, U2, . . . , U2t1 , V2, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of the cycles C2t1 and C2t2, and A = t1∑ i=1 |Ui||Ui+t1 |+ 2t2∑ j=2 |Ut1+1||Vj |+ |Ut1+1||Vt2+1|, B = t2∑ j=1 |Vj ||Vj+t2 |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|, and C = t1∑ i=1 |Ui||Ui+t1 |+ t2∑ j=1 |Vj ||Vj+t2 |+ 2t2∑ j=2 |Ut1+1||Vj |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. Proof. It’s enough to show the bounds of Wiener index in terms of the Gutman index and use Lemma 2 and the proof of Theorem 1. Now, W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ {u,v}⊆V (G) d(u, v) · dudv dudv < 1 δ2 Gut(G). Moreover, W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ {u,v}⊆V (G) d(u, v) · dudv dudv > 1 ∆2 Gut(G). R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 12 of 20 The strict inequalities in the above result are due to the fact that there exist u, v ∈ V (G) such that dudv > δ2. The following results uses Theorem 2 and the proof of Theorem 3. Corollary 1. Let G be a simple spirocyclic graph with minimum degree δ = 2 and with even cycles C2t1 and C2t2, for some natural numbers t1, t2 ≥ 2. Then 1 16 Gut(G) + 2(t1 + t2) 2 < Wg(G) < 1 4 Gut(G) + 2(t1 + t2) 2. 7. Bounds of g-Wiener Index with Schultz Index The following result establishes the relation between geodetic-Wiener and Schultz [19] indices for simple spirocyclic graphs containing even cycles. Theorem 4. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2. Then 1 2∆ S(G) +At1 +Bt2 < Wg(G) < 1 2δ S(G) + Cdiam(G), where {U1 = V1, U2, . . . , U2t1 , V2, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of the cycles C2t1 and C2t2, and A = t1∑ i=1 |Ui||Ui+t1 |+ 2t2∑ j=2 |Ut1+1||Vj |+ |Ut1+1||Vt2+1|, B = t2∑ j=1 |Vj ||Vj+t2 |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|, and C = t1∑ i=1 |Ui||Ui+t1 |+ t2∑ j=1 |Vj ||Vj+t2 |+ 2t2∑ j=2 |Ut1+1||Vj |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. Proof. We will establish the bounds of Wiener index in terms of Schultz index and use Lemma 2 together with the proof of Theorem 1. W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ u,v∈V (G) d(u, v) · du + dv du + dv < 1 2δ S(G). R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 13 of 20 Moreover, W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ {u,v}⊆V (G) d(u, v) · du + dv du + dv > 1 2∆ S(G). Using Theorem 2 and the proof of Theorem 4, the following is immediate. The strict inequalities in the above result are due to the fact that there exist u, v ∈ V (G) such that du + dv > 2δ. Corollary 2. Let G be a simple spirocyclic graph with minimum degree δ = 2 and with even cycles C2t1 and C2t2, for some natural numbers t1, t2 ≥ 2. Then 1 8 S(G) + 2(t1 + t2) 2 < Wg(G) < 1 4 S(G) + 2(t1 + t2) 2. 8. Bounds of g-Wiener Index with Hyper-Wiener Index The following result establishes the bounds of geodetic-Wiener index in terms of hyper- Wiener index as defined in [16] for simple spirocyclic graphs with even cycles. Theorem 5. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2. Then WW (G) +At1 +Bt2 < Wg(G) < 2 diam(G) + 1 WW (G) + Cdiam(G) where {U1 = V1, U2, . . . , U2t1 , V2, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of the cycles C2t1 and C2t2, and A = t1∑ i=1 |Ui||Ui+t1 |+ 2t2∑ j=2 |Ut1+1||Vj |+ |Ut1+1||Vt2+1|, B = t2∑ j=1 |Vj ||Vj+t2 |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|, and C = t1∑ i=1 |Ui||Ui+t1 |+ t2∑ j=1 |Vj ||Vj+t2 |+ 2t2∑ j=2 |Ut1+1||Vj |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 14 of 20 Proof. The proof is established by taking the bounds of Wiener index in terms of Hyper-Wiener index and using Lemma 2 together with the proof of Theorem 1. W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ {u,v}⊆V (G) d(u, v) · d(u, v) + 1 d(u, v) + 1 < 2 diam(G) + 1 WW (G). Moreover, W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ {u,v}⊆V (G) d(u, v) · d(u, v) + 1 d(u, v) + 1 > WW (G) The proof is complete. An immediate result for simple spirocyclic graph with minimum degree δ = 2, which uses Theorem 2 and the proof of Theorem 5, is shown below. Corollary 3. Let G be a simple spirocyclic graph with minimum degree δ = 2 and with even cycles C2t1 and C2t2, for some natural numbers t1, t2 ≥ 2. Then 2 t1 + t2 + 1 WW (G) + 2(t1 + t2) 2 < Wg(G) < WW (G) + 2(t1 + t2) 2. 9. Bounds of g-Wiener Index with Harary Index The following result establishes the relation between g-Wiener index and Harary index for simple spirocyclic graphs. Theorem 6. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2, without a common edge. Then H(G) +At1 +Bt2 < Wg(G) < (diam(G))2H(G) + Cdiam(G) where {U1 = V1, U2, . . . , U2t1 , V2, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of the cycles C2t1 and C2t2, and A = t1∑ i=1 |Ui||Ui+t1 |+ 2t2∑ j=2 |Ut1+1||Vj |+ |Ut1+1||Vt2+1|, R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 15 of 20 B = t2∑ j=1 |Vj ||Vj+t2 |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|, and C = t1∑ i=1 |Ui||Ui+t1 |+ t2∑ j=1 |Vj ||Vj+t2 |+ 2t2∑ j=2 |Ut1+1||Vj |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. Proof. It suffices to show the bounds of Wiener index in terms Harary index and use Lemma 2 together with Theorem 1. W (G) = ∑ {u,v}∈V (G) d(u, v) = diam(G) ∑ u,v⊆V (G) d(u, v) d(u, v) < diam(G)2H(G). Moreover, W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ {u,v}⊆V (G) 1 d(u, v) > H(G). The proof is complete. For simple spirocyclic graphs with minimum degree δ = 2, the following bounds of Harary index follows by using Theorem 2 and the proof of Theorem 6. Corollary 4. Let G be a simple spirocyclic graph with minimum degree δ = 2 and with even cycles C2t1 and C2t2, for some natural numbers t1, t2 ≥ 2. Then (t1 + t2) 2H(G) + 2(t1 + t2) 2 < Wg(G) < H(G) + 2(t1 + t2) 2. 10. Bounds of g-Wiener Index with Additively Weighted Harary Index The following result establishes bounds of g-Wiener index in terms of additively weighted Harary index as defined in [12] for simple spirocyclic graph G containing even cycles. Theorem 7. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 16 of 20 and C2t2 = [u1 = v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2. Then 1 2∆ HA(G) +At1 +Bt2 < Wg(G) < (diam(G))2 2δ HA(G) + Cdiam(G) where {U1 = V1, U2, . . . , U2t1 , V2, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of the cycles C2t1 and C2t2, and A = t1∑ i=1 |Ui||Ui+t1 |+ 2t2∑ j=2 |Ut1+1||Vj |+ |Ut1+1||Vt2+1|, B = t2∑ j=1 |Vj ||Vj+t2 |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|, and C = t1∑ i=1 |Ui||Ui+t1 |+ t2∑ j=1 |Vj ||Vj+t2 |+ 2t2∑ j=2 |Ut1+1||Vj |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. Proof. We use Lemma 2 and the proof of Theorem 1 to simplify the proof. In this case, it is enough to show the bounds of Wiener index in terms of Additively Weighted Harary index. W (G) = ∑ {u,v}⊆V (G) d(u, v) = diam(G) ∑ {u,v}⊆V (G) du + dv d(u, v) · d(u, v) du + dv < (diam(G))2 2δ HA(G). Moreover, W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ {u,v}⊆V (G) 1 d(u, v) · du + dv du + dv > 1 2∆ HA(G). This completes the proof. The next corollary establishes the bounds of geodetic-Wiener index with additively Weighted Harary index for simple spirocyclic graphs with minimum degree δ = 2 by R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 17 of 20 applying Theorem 2 and the proof of Theorem 7. Corollary 5. Let G be a simple spirocyclic graph with minimum degree δ = 2 and with even cycles C2t1 and C2t2, for some natural numbers t1, t2 ≥ 2. Then (t1 + t2) 2 4 HA(G) + 2(t1 + t2) 2 < Wg(G) < 1 8 HA(G) + 2(t1 + t2) 2. 11. Bounds of g-Wiener Index with Multiplicatively Weighted Harary Index The following result establishes bounds of g-Wiener index in terms of the multiplica- tively weighted Harary index as defined in [12] for classes of simple spirocyclic graphs containing even cycles. Theorem 8. Let G be a simple spirocyclic graph with even cycles C2t1 = [u1, u2, . . . , u2t1 ] and C2t2 = [v1, v2, . . . , v2t2 ], for some natural numbers t1, t2 ≥ 2. Then 1 ∆2 HM (G) +At1 +Bt2 < Wg(G) < (diam(G))2 δ2 HM (G) + Cdiam(G) where {U1 = V1, U2, . . . , U2t1 , V2, . . . , V2t2} is the tree-partition of V (G) with respect to the projection of V (G) onto the union of the cycles C2t1 and C2t2, and A = t1∑ i=1 |Ui||Ui+t1 |+ 2t2∑ j=2 |Ut1+1||Vj |+ |Ut1+1||Vt2+1|, B = t2∑ j=1 |Vj ||Vj+t2 |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|, and C = t1∑ i=1 |Ui||Ui+t1 |+ t2∑ j=1 |Vj ||Vj+t2 |+ 2t2∑ j=2 |Ut1+1||Vj |+ 2t1∑ i=2 |Vt2+1||Ui|+ |Ut1+1||Vt2+1|. Proof. It suffices to establish the bounds of Wiener index in terms of Multiplica- tively Weighted Harary and use Lemma 2 together with Theorem 1. Note that in simple spirocyclic graphs, δ < ∆. Hence, W (G) = ∑ {u,v}⊆V (G) d(u, v) ≤ diam(G) ∑ {u,v}⊆V (G) dudv d(u, v) · d(u, v) dudv < [diam(G)]2 δ2 HM (G). R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 18 of 20 Moreover, W (G) = ∑ {u,v}⊆V (G) d(u, v) = ∑ {u,v}⊆V (G) 1 d(u, v) · dudv dudv > 1 ∆2 HM (G). The proof is complete. Applying Theorem 2 and the proof of Theorem 8 on the bounds of Wiener index with multiplicatively weighted Harary index yields the following corollary. Corollary 6. Let G be a simple spirocyclic graph with minimum degree δ = 2 and with even cycles C2t1 and C2t2, for some natural numbers t1, t2 ≥ 2. Then 1 16 HM (G) + 2(t1 + t2) 2 < Wg(G) < (t1 + t2) 2 4 HM (G) + 2(t1 + t2) 2. In Theorems 5-8, the strict inequalities are due to the fact that there exist u, v ∈ V (G) such that d(u, v) < diam(G). In particular, if uv ∈ V (G), then d(u, v) = 1 < diam(G). 12. Conclusion In this work, we found some bounds for geodetic-Wiener index in terms of other distance-based topological indices for simple spirocyclic graphs with even cycles by de- composing the structure into subtrees using the concept of projection of V (G) onto the even cycles. Moreover, we have shown that the geodetic-Wiener index of spirocyclic graphs without pendant vertices is equal to the Wiener index plus twice the square of the sum of the diameters of the cycles. The g-Wiener index captures more structural information compared to standard Wiener index. A concrete example is a spirocyclic compound with even connecting cycles, such as the spirobicyclohexane. Consider graphs with k even cycles where k ≥ 3. This will increase the number of geodesics between some vertex pairs in the graph. Also, when an odd cycle is attached to the even cycle, the projection of some vertices to the even cycle may not be unique. This will be another interesting and challenging problem that could be considered. Further- more, evaluation of g-Wiener index of graphs resulting from some unary and binary graph operations are potential problems that could be considered for further investigations by expressing the g-Wiener index of the graph resulting from the graph operations in terms of the g-Wiener index of the original graph being considered. R. G. Artes Jr. et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6212 19 of 20 Acknowledgements The first author is supported by the MSU-TCTO APDP Grant. References [1] Danishuddin and A. U. Khan. Descriptors and their selection methods in qsar anal- ysis: paradigm for drug design. Drug Discovery Today, 8(8):1291–1302, 2016. [2] H. Wiener. Structural determination of paraffin boiling points. Journal of the Amer- ican Chemical Society, 69(1):17–20, 1947. [3] M. Randić. Novel molecular descriptor for structure-property studies. Chemical Physics Letters, 211:478–483, 1993. [4] I. Gutman. Degree-based topological indices. 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Preprint, 2025. https://chembridge.com/targeted-and-specialty-libraries/spirocycles/ https://chembridge.com/targeted-and-specialty-libraries/spirocycles/ https://molview.org/ https://www.lookchem.com/casno/180-43-8.html https://www.lookchem.com/casno/180-43-8.html Introduction Spirocyclic Compounds and Applications Geodetic-Wiener Index Formulation The Partitioning Bounds of g-Wiener Index with Wiener Index Bounds of g-Wiener Index with Gutman Index Bounds of g-Wiener Index with Schultz Index Bounds of g-Wiener Index with Hyper-Wiener Index Bounds of g-Wiener Index with Harary Index Bounds of g-Wiener Index with Additively Weighted Harary Index Bounds of g-Wiener Index with Multiplicatively Weighted Harary Index Conclusion