Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 71 https://internationalpubls.com Some Reverse Topological Indices of Comet and Double Comet Graphs John Rafael M. Antalan1*, Richard P. Tagle2 1Department of Mathematics and Physics, College of Science Central Luzon State University 2Science City of Muñoz, Nueva Ecija, 3120, Philippines jrantalan@clsu.edu.ph Article History: Received: 22-07-2024 Revised: 10-09-2024 Accepted: 29-09-2024 Abstract Introduction: Chemical Graph Theory or CGT for short, is a transdisciplinary field of Mathematics wherein graphs are used to represent chemical compounds. Under this representation, the atoms of a chemical compound are expressed as vertices, while the bonds connecting these atoms are expressed as edges. Once the graph representation of a chemical compound has been determined, graph-theoretic techniques can now be used to determine various topological indices associated with the chemical compound. Topological indices are invariants of a graph that have predictive power for the chemical properties of a chemical compound. In CGT, trees, being connected and acyclic, have been an essential class of graphs. This is because, most compounds have acyclic molecular structures. In the 2023 study by Gowtham and Husin, various reverse topological indices of a family of trees called bistar graphs have been determined. Objectives: Motivated by the work of Gowtham and Husin, we determine some reverse topological indices of another family of trees called comets and double comets. Double comets are natural extension of bistar graphs. We also give a certain computational application of our results on double comet graphs. Finally, we investigate the relationship between the reverse topological indices of bistar graphs and double comets. Methods: Several important graph-theoretic concepts were used to analyze some important properties of comets and double comets, leading to the computation of their reverse vertex degree based topological indices. Results: The following reverse vertex degree-based topological indices for comets and double comets were determined: Reverse sum-connectivity, First reverse Zagreb, Second reverse Zagreb, Reverse arithmetic-geometric, Reverse geometric-arithmetic, Reverse Sombor, and Reverse Nirmala. A computational application of the results to a certain chemical compound (2,2,4,4-Tetramethylpentane) or C9H20 were also demonstrated. Conclusions: The results of the paper provided an extension to an existing result on the reverse vertex degree-based topological indices of bistar graphs by considering double comet graphs. Several topological reverse vertex degree-based topological indices were also computed for comet graphs. The computed topological indices were also applied in determining some invariants of a certain chemical compound. For future studies, we recommend that reverse vertex degree based topological indices of other family of trees will be considered. Keywords: molecular graph, topological index, reverse vertex degree, comet, double comet AMS Classification Numbers: 05C05, 05C07, 05C09 1. Introduction mailto:jrantalan@clsu.edu.ph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 72 https://internationalpubls.com In this paper, the graphs that we will consider are all simple, connected, and finite. Also, throughout this paper, we follow the common notation 𝑉(𝐺) and 𝐸(𝐺) for the vertex set and edge set of a graph 𝐺, respectively. We also have adapted the graph theory notations used by Wagner and Wang in [27], and refer to the same text, for any reader who needs to recall some graph-theoretic concepts that were mentioned, but not explicitly discussed in detail in this paper. In CGT, the atoms of a chemical compound and the bonds connecting them are expressed respectively as vertices and edges of a graph, thereby obtaining its molecular graph. Under this representation, the vertices corresponding to hydrogen atoms are usually removed since the valency or degree of Hydrogen is one. For example, the ball and stick chemical structure of 2,2,4,4-Tetramethylpentane (C9H20) in Figure 1a has a corresponding molecular graph shown in Figure 1b. Figure 1: (a)The ball and stick structure of 2,2,4,4-Tetramethylpentane C9H20 taken from PubChem [1]. (b) The molecular graph of 2,2,4,4-Tetramethylpentane. Once the molecular graph of a chemical compound has been obtained, one can now use graph theoretic concepts and techniques to determine various topological indices associated with the compound. A topological index is a computed numerical value associated with the molecular graph that has predictive power for the chemical properties of the chemical compound. Topological indices may be classified according to the graph property in which they are defined. It is said to be distance-based, if it is derived based on the concept of distance between each pair of vertices in the graph. On the other hand, a topological index is vertex degree-based, if it is defined based on the degree of the vertices in the graph. One popular topological index based on distance is the Wiener index denoted by W(·). For a graph 𝐺, its Wiener index is defined as “W(G) = ∑ d(u,v)”, {u,v}⊆V(G) wherein the notation d(u,v) refers to the distance between the vertices u and v in 𝐺. This index was named after Harry Wiener, a chemist who proposed this concept in 1947 [29]. Since the publication of this work by Wiener, various studies related to this index have been conducted with the recent works provided in [4, 5, 7, 6, 8, 9, 11, 20, 21, 31]. An example of a topological index based on vertex degree is the Randic index and its generalization denoted by R(·) and Rα(·) respectively, where α ≠ 0. If 𝐺 is a graph, then its generalized Randic index is given by “Rα(G) = ∑ [degG(u)degG(v)]α”, uv∈E(G) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 73 https://internationalpubls.com wherein degG(u) and degG(v) refers to the degrees of vertices u and v in G respectively. For α = − 1 2 , we have the “Randic index” while for α = 1, we have the “second Zagreb index”. The Randic index was named after Milan Randic who proposed the index in [25] in the year 1975. The second Zagreb index, on the other hand, was introduced by Ivan Gutman and Nenad Trinajstic in 1972 and first appeared in [13]. These two vertex degree-based indices have been constantly considered as a topic of study, and are being used in the Quantitative Structure-Property Relationships (QSPR) modeling. Some recent use of Randic and the second Zagreb index and their variations for QSPR modeling can be shown in [3, 14, 15, 16, 17, 23, 24, 30, 33, 35]. In this study however, we will consider another type of topological indices which are dependent on the “reverse vertex-degree” of a vertex in a graph. Suppose 𝐺 is a graph with maximum vertex degree ∆(G). For any vertex u in G, its reverse vertex degree, denoted by ru is the quantity ∆(G)−degG(u)+1, that is, “ru = ∆(G) − degG(u)+1”. This concept was introduced by Kulli in 2018, where he computed the reverse Zagreb and reverse hyper-Zagreb index of rhombus silicate networks [18]. Since then, various research works on the determination of these types of indices for graphs and chemical materials have been conducted. These works can be viewed in [2, 19, 22, 26, 28, 32, 34, 36] and were discussed by Gowtham and Husin in their recent research article [12], which is the motivational work for this research study. Gowtham and Husin computed some reverse vertex-degree-based topological indices of bistar graphs which is an example of a tree [12]. Trees, being connected and acyclic, have been an essential class of graphs in CGT. For this study, we will determine some reverse vertex degree-based topological indices of other families of trees called comets and double comets. 2. Preliminary Concepts and Results We will briefly discuss the recent research work of Gowtham and Husin regarding some reverse vertex degree-based topological indices of bistar graphs [12]. Moreover, essential preliminary results on comets and double comets will also be discussed which will be used in the results section. 2.1 Gowtham and Husin’s Results for Bistar Graphs A vertex u ∈ V(G) is said to be a leaf if degG(u) = 1. A star of order n+1, denote by K1,n is a tree with n number of leaves and a central vertex of degree n. Presented in Figure 2 are the stars K1,n and K1,m respectively. Figure 2: (a) The star K1,n. (b) The star K1,m. The bistar graph denoted by B(n;m) is a graph obtained by connecting the centers of two star graphs with n and m number of leaves respectively by an edge. The graph B(n;m) is depicted in Figure 3. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 74 https://internationalpubls.com Figure 3: The bistar graph B(n;m). Gowtham and Husin calculated the following reverse vertex degree-based topological indices of B(n;m): Reverse sum-connectivity (RSCI(·)), First reverse Zagreb (RM1(·)), Second reverse Zagreb (RM2(·)), Reverse arithmetic-geometric (RAG(·)), Reverse geometric-arithmetic (RGA(·)), Reverse Sombor (RSO(·)), and Reverse Nirmala (RN(·)) in [12]. Their results are summarized in the table below. To eliminate cases, we introduce the notations M = max{m,n} and µ = min{m,n}. Table 1: Various reverse degree-based topological indices of B(n;m) obtained by Gowtham and Husin. Motivated by the above results from Gowtham and Husin, we determine the same set of topological indices for comets and double comets in this paper. 2.2. Reverse Vertex Degree of Vertices of Comets and Double Comets As mentioned in the previous subsection, this paper will investigate several reverse vertex degree- based topological indices of comets and double comets. In this subsection, we will discuss essential preliminary results on comets and double comets which are needed for the discussion of main results. We begin by providing the definition of a path. A path with n number of vertices, denoted by Pn is the unique tree with only two leaves. The non-leaf vertices of a path are called internal vertices. In general, internal vertices are vertices of degree ≥ 2. The path P4 with leaf vertices v1 and v4 and internal vertices v2 and v3 is presented in Figure 4. Figure 4: The path P4 with v1, v4 as leaves and v2, v3 as internal vertices. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 75 https://internationalpubls.com We are now at the position to define comet graphs. The comet Cm,n is the tree obtained from path Pm by appending n > 1 edges to one end of Pm. The end vertex of Pm appended by n number of edges is called the branching vertex of Cm,n. In general, a branching vertex is a vertex of degree ≥ 3. We remark that the notation we used in denoting comet graphs, and later for double comets are the one used by Dogan Durgun and Toprakkaya in [10]. The comet C4,5 with branching vertex vb is depicted in Figure 5a while the comet Cm,n is depicted in Figure 5b. Figure 5: (a) The comet C4,5 with branching vertex vb. (b) The comet Cm,n. On the other hand, the double comet DC(m,m1,m2) is the tree obtained from path Pm by appending m1 > 1 edges to one end of Pm and m2 > 1 edges on the other end of Pm. In this case, the ends of Pm are the branching vertices of DC(m,m1,m2). The double comet DC(4,3,5) is shown in Figure 6a, while the double comet DC(m,m1,m2) is depicted in Figure 6b. Figure 6: (a) The double comet DC(4,3,5). (b) The double comet DC(m,m1,m2). Remark 2.1. Before we continue, we remark that although based on definition, a branching vertex is an internal vertex, we reserve the term “internal vertex” for vertices that are internal but not branching in this paper. Now that we have already discussed the comet and double comet graphs, we are now ready to present results concerning the reverse vertex degree of their vertices. Proposition 2.2. If u is a vertex in V(Cm,n) then Proof. We begin by noting that for G =Cm,n, we have ∆(G) = n+1. Let u ∈V(G). Since u is a vertex in V(G), then u is either a leaf with degG(u) = 1, an internal vertex with degG(u) = 2, or a branching vertex with degG(u) = n+1. If u is a leaf, then ru = ∆(G) − degG(u) + 1 = (n +1) −1+1 = n +1. If u is an internal vertex, then ru = ∆(G) − degG(u) + 1 = (n + 1) – 2 + 1 = n. Finally, if u is a branching vertex, then ru = ∆(G) − degG(u) +1 = (n + 1) − (n + 1) + 1 = 1. This proves Proposition 2.2. We now end the section by stating and proving the next proposition. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 76 https://internationalpubls.com Proposition 2.3. Let M = max{m1,m2} and µ = min{m1,m2}. If u is a vertex in V(DC(m,m1,m2)) then Proof. We begin by noting that for G = DC(m,m1,m2), we have ∆(G) = M +1. Let u ∈ V(G). Since u is a vertex in V(G), then u is either a leaf with degG(u) = 1, an internal vertex with degG(u) = 2, a branching vertex with degG(u) = µ +1, and a branching vertex with degG(u) = M +1. If u is a leaf, then ru = ∆(G) − degG(u) + 1 = (M +1)−1+1 = M +1. If u is an internal vertex, then ru =∆(G) − degG(u)+1 =(M + 1) – 2 +1 = M. If u is a branching vertex with degG(u)= µ +1, then ru = ∆(G) − degG(u)+1 = (M + 1) − (µ +1) + 1 = M − µ +1. Finally, if u is a branching vertex with degG(u) = M + 1, then ru = ∆(G) − degG(u)+ 1 = (M + 1) − (M + 1) + 1 = 1. This proves Proposition 2.3. 3. Several Reverse Vertex Degree Indices of Comets and Double Comets 3.1. Results for Comet Graphs We now present our results for comet graphs. Throughout this subsection, we denote the comet graph Cm,n by G. Theorem 3.1. Let G =Cm,n. The “reverse sum-connectivity index” of G denoted by RSCI(G) is given by Proof. We start by noting that the graph G =Cm,n has m – 1 + n edges with the following classification based on their end vertices: 1 leaf-internal edge, m − 3 internal-internal edges, 1 internal-branching edge, and n branching-internal edges. Combining these information with equation (1) in Proposition 2.2 and the definition of the reverse sum-connectivity index, we arrive at Theorem 3.2. Let G =Cm,n. The “first reverse Zagreb index” of G denoted by RM1(G) is given by RM1(G) = (n + 1)3 + mn 2− 2n2 + 1. Proof. We start by noting that the graph G =Cm,n has m + n vertices with the following classification: n+1 leaf vertices, m −2 internal vertices, and 1 branching vertex. Combining these information with equation (1) in Proposition 2.2 and the definition of the first reverse Zagreb index, we arrive at Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 77 https://internationalpubls.com Theorem 3.3. Let G =Cm,n. The “second reverse Zagreb index” of G denoted by RM2(G) is given by RM2(G) = n(mn – n + 3). Proof. The proof is similar to the proof of Theorem 3.1. Theorem 3.4. Let G = Cm,n. The “reverse arithmetic-geometric index” of G denoted by RAG(G) is given by Proof. We start by noting that the graph G =Cm,n has m−1+n edges with the following classification based on their end vertices: 1 leaf-internal edge, m−3 internal-internal edges, 1 internal-branching edge, and n branching-internal edges. Combining these information with equation (1) in Proposition 2.2 and the definition of the reverse arithmetic-geometric index, we arrive at Theorem 3.5. Let G = Cm,n. The “reverse geometric-arithmetic index” of G denoted by RGA(G) is given by Proof. The proof is similar to the proof of Theorem 3.4. Theorem 3.6. Let G =Cm,n. The “reverse Sombor index” of G denoted by RSO(G) is given by Proof. As with the previous proofs, we start with the fact that the graph G =Cm,n has m – 1 + n edges with the following classification based on their end vertices: 1 leaf-internal edge, m−3 internal-internal edges, 1 internal-branching edge, and n branching-internal edges. Combining these information with equation (1) in Proposition 2.2 and the definition of the reverse Sombor index, we arrive at Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 78 https://internationalpubls.com Theorem 3.7. Let G =Cm,n. The “reverse Nirmala index” of G denoted by RN(G) is given by Proof. The proof is similar to the proof of Theorem 3.6. 3.2. Results for Double Comet Graphs Next, we present our results for double comet graphs. Throughout this subsection, we denote the double comet graph DC(m,m1,m2) by G, the maximum of {m1,m2} by M, and the minimum of {m1,m2} by µ. Theorem 3.8. Let G = DC(m,m1,m2). Denote by M = max{m1,m2} and µ = min{m1,m2}. The “reverse sum-connectivity index” of G denoted by RSCI(G) is given by Proof. For the proof of this result, we refer the readers to the proof provided in Theorem 3.10, since they are similar in nature. Theorem 3.9. Let G = DC(m,m1,m2). Denote by M = max{m1,m2} and µ = min{m1,m2}. The “first reverse Zagreb index” of G denoted by RM1(G) is given by Proof. We start by noting that the graph G = DC(m,m1,m2) has m + m1 +m2 = m + µ + M vertices with the following classification: m1 + m2 = µ + M leaf vertices, m − 2 internal vertices, 1 branching vertex of degree µ +1, and 1 branching vertex of degree M +1. Combining these information with equation (2) in Proposition 2.3 and the definition of the first reverse Zagreb index, we arrive at Theorem 3.10. Let G = DC(m,m1,m2). Denote by M = max{m1,m2} and µ = min{m1,m2}. The“second reverse Zagreb index” of G denoted by RM2(G) is given by Proof. We start by noting that the graph G = DC(m,m1,m2) has (m −1) + m1 + m2 = (m −1)+µ +M edges with the following classification based on their end vertices: µ +M leaf-branching (branching-leaf) edge, 2 internal-branching (branching-internal) edge, and m−3 internal-internal edges. Before we proceed, we emphasize that in the µ +M leaf-branching (branching-leaf) edge type, there are µ number of edges whose one end is the vertex with degree µ +1 and M number of edges whose one end is the vertex with degree M +1. Also in the 2 internal-branching (branching-internal) edge type, 1 edge has Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 79 https://internationalpubls.com the vertex of degree µ +1 as one end vertex, while the other edge has the vertex of degree M +1 as one end vertex. Combining these information with equation (2) in Proposition 2.3 and the definition of the second reverse Zagreb index, we arrive at Theorem 3.11. Let G = DC(m,m1,m2). Denote by M = max{m1,m2} and µ = min{m1,m2}. The “reverse arithmetic-geometric index” of G denoted by RAG(G) is given by Proof. The proof is similar to the proof provided in Theorem 3.12. Theorem 3.12. Let G = DC(m,m1,m2). Denote by M = max{m1,m2} and µ = min{m1,m2}. The “reverse geometric-arithmetic index” of G denoted by RGA(G) is given by Proof. Similar to what we did earlier, we start by noting that the graph G = DC(m, m1, m2) has (m− 1) + m1 + m2 = (m −1) + µ +M edges with the following classification based on their end vertices: µ+M leaf-branching (branching-leaf) edge, 2 internal-branching (branching-internal) edge, and m−3 internal-internal edges. Again, we emphasize that in the µ +M leaf-branching (branching-leaf) edge type, there are µ number of edges whose one end is the vertex with degree µ +1 and M number of edges whose one end is the vertex with degree M +1. Also in the 2 internal-branching (branching- internal) edge type, 1 edge has the vertex of degree µ + 1 as one end vertex, while the other edge has the vertex of degree M +1 as one end vertex. Combining these information with equation (2) in Proposition 2.3 and the definition of the reverse geometric-arithmetic index, we arrive at Simplifying the above displayed equation will give us This proves the theorem. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 80 https://internationalpubls.com Theorem 3.13. Let G = DC(m,m1,m2). Denote by M = max{m1,m2} and µ = min{m1,m2}. The “reverse Sombor index” of G denoted by RSO(G) is given by Proof. The proof of this theorem is similar to the proof of Theorem 3.14. Theorem 3.14. Let G = DC(m,m1,m2). Denote by M = max{m1,m2} and µ = min{m1,m2}. The “reverse Nirmala index” of G denoted by RN(G) is given by Proof. Again, we start by noting that the graph G = DC(m,m1,m2) has (m −1) + m1 + m2 = (m − 1) +µ + M edges with the following classification based on their end vertices: µ +M leaf-branching (branching-leaf) edge, 2 internal-branching (branching-internal) edge, and m − 3 internal-internal edges. Moreover, we emphasize that in the µ+M leaf-branching (branching-leaf) edge type, there are µ number of edges whose one end is the vertex with degree µ +1 and M number of edges whose one end is the vertex with degree M +1. Also in the 2 internal-branching (branching-internal) edge type, 1 edge has the vertex of degree µ +1 as one end vertex, while the other edge has the vertex of degree M +1 as one end vertex. Combining these information with equation (2) in Proposition 2.3 and the definition of the reverse Nirmala index, we arrive at 3.3 Reverse Vertex Degree Indices of 2,2,4,4-Tetramethylpentane For this subsection, we illustrate a particular computational application of Theorems 3.8 - 3.14 obtained in the previous section. Observe that the molecule 2,2,4,4-Tetramethylpentane in the introductory section has DC(3,3,3) as a molecular graph. Hence, we have the following immediate results. Corollary 3.15. Let G denote the molecular graph of 2,2,4,4-Tetramethylpentane, then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 81 https://internationalpubls.com 3.4 Relationship Between the Results for Bistar and for Double Comet This subsection investigates the relationship between the reverse vertex degree-based topological indices of bistars and double comets. The motivation for this subsection is the fact that the bistar B(m1 ,m2 ) is the double comet DC(2 m1 ,m2). Hence, we can obtain DC(m, m1 ,m2) from DC(2, m1 ,m2) by expanding P2 to Pm. Specifically, by adding vertices and edges to P2 to become Pm. The results in this subsection are obtained by calculating the difference between a particular reverse topological index of the double comet and bistar. For instance, if the concerned topological index is RM1, then we calculate RM1(DC(m,m1,m2))−RM1(B(m1,m2)). The computation is of course based on the results of Gowtham and Husin summarized in subsection 2.1, and the results of this paper provided in subsection 3.2. Theorem 3.16. Let G1 =DC(m,m1,m2) and G2 =B(m1,m2). If M =max{m1,m2} and µ =min{m1,m2}, then Theorem 3.17. Let G1 =DC(m,m1,m2) and G2 =B(m1,m2). If M =max{m1,m2} and µ =min{m1,m2}, then Theorem 3.18. Let G1 =DC(m,m1,m2) and G2 =B(m1,m2). If M =max{m1,m2} and µ =min{m1,m2}, then Theorem 3.19. Let G1 =DC(m,m1,m2) and G2 =B(m1,m2). If M =max{m1,m2} and µ =min{m1,m2}, then Theorem 3.20. Let G1 =DC(m,m1,m2) and G2 =B(m1,m2). If M =max{m1,m2} and µ =min{m1,m2}, then Theorem 3.21. Let G1 =DC(m,m1,m2) and G2 =B(m1,m2). If M =max{m1,m2} and µ =min{m1,m2}, then Theorem 3.22. Let G1 =DC(m,m1,m2) and G2 =B(m1,m2). If M =max{m1,m2} and µ =min{m1,m2}, then 4. Conclusion and Future Work In this paper, the results obtained in Theorem 3.1 - Theorem 3.7 show that we were able to compute for the Reverse sum-connectivity (RSCI(·)), First reverse Zagreb (RM1(·)), Second reverse Zagreb Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 82 https://internationalpubls.com (RM2(·)), Reverse arithmetic-geometric (RAG(·)), Reverse geometric-arithmetic (RGA(·)), Reverse Sombor (RSO(·)), and Reverse Nirmala (RN(·)) of comet graphs. The same reverse vertex degree- based topological indices were calculated for double comet graphs and were presented in Theorem 3.8 - Theorem 3.14. We also have presented a computational application of our results to a specific molecule. 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