Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 123 https://internationalpubls.com Bounds on The Reduced Sombor Index of Graphs S. NagarajanΒΉ, B. AswiniΒ² ΒΉDepartment of Mathematics, Kongu Arts and Science College (Autonomous), Erode, Tamilnadu-638 107 profnagarajan.s@gmail.com Β²School of Mathematics, A.V.P. College of Arts and Science, Tirupur-641 652. aswiniprasaad@gmail.com Article History: Received: 28-07-2024 Revised: 07-09-2024 Accepted: 17-09-2024 Abstract: The reduced Sombor index is a modified version of the very famous Sombor Index for a graph 𝐺. In this article, the reduced Sombor index is studied on various classes of graphs and novel results on the bounds of the reduced Sombor index of graphs are obtained. The graphs with minimum reduced Sombor index are studied, and the graphs achieving certain bounds are found. Keywords: Reduced Sombor index; topological index; graph invariant; trees; extremal problem; characterization. 1. Introduction By a graph G, in this article, we mean an ordered pair (𝑉𝐺 , 𝐸𝐺), and the members of the sets 𝑉𝐺 and 𝐸𝐺 respectively are the vertices and edges of the graph. The set of vertices that are adjacent to a vertex 𝑒 in 𝐺 is denoted as 𝑁𝐺(𝑒) and called by β€œthe open neighbourhood” of 𝑒 in 𝐺. The term β€œclosed neighbourhood” is 𝑁𝐺[𝑣] = 𝑁𝐺(𝑣) βˆͺ {𝑣} and by a (𝑣, 𝑀)-Path 𝑣1𝑣2 . . . 𝑀 is a sequence of distinct members of the set 𝑉𝐺 and the vertices 𝑣, 𝑀 are usually known as the origin and the terminus of the path 𝑃 respectively. The concept of distance between any two vertices π‘₯, 𝑦 ∈ 𝑉𝐺 is usually defined as the length of the smallest (π‘₯, 𝑦)-path that exists in 𝐺. If 𝑑𝐺(𝑒) = 1, then v is a pendant vertex and it is adjacent to a unique vertex in 𝐺, say 𝑒 which is called a support vertex. For more on graphs and related works, the reader is referred to [1– 3]. The topological indices (also known as graph invariants) play a major role in the chemical graph theory because they are used to analyze the behaviour of the molecule structures and their inter- relationships. There are numerous topological indices available in the literature; a few of them are the Sombor index, Zagreb index, and so on. The topological indices were defined with minor and major modifications in the past and several classes of topological indices are available for the Sombor index and Zagreb index. Given a graph, the Sombor (SO) index is defined (by Gutman[4]) as 𝑆𝑂(𝐺) = βˆ‘ βˆšπ‘‘πΊ(𝑒)2 + 𝑑𝐺(𝑣)2 π‘ˆ,π‘‰βˆˆπΈπΊ The Sombor index, in recent years, received numerous attentions from academics and researchers throughout the globe [8–11]. For some recent surveys in Sombor index, one can refer to the articles [6, 7]. Chemical applications have been carried out in the articles [12, 13]. For various results and versions of Sombor index, one can refer [5, 14–17]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 124 https://internationalpubls.com The reduced Sombor index is defined as 𝑆𝑂(𝐺) = βˆ‘ √(𝑑𝐺(𝑒) βˆ’ 1)2 + (𝑑𝐺(𝑣) βˆ’ 1)2 𝑒,π‘£βˆˆπΈπΊ The reduced Sombor index is a recently introduced term and some of the works can be found in [18, 19]. The reduced Sombor index, for a wide collection of graphs, is studied in this article and characterized the graphs with minimum reduced Sombor index of connected and disconnected graphs. 2. Objectives In this paper, the emphasis is on identifying the limits of the reduced Sombor index, a newly introduced topological measure applied in the exploration of chemical graph theory and related disciplines. The Sombor index is derived from the degrees of nodes in a graph, and the reduced Sombor index further elaborates on this idea. The researchers investigate the smallest and largest values of the reduced Sombor index over a wide range of graphs. By examining various graph configurations, they are able to define the specific types of graphs that reach these boundary values. This study enhances the understanding of how graph structure impacts the reduced Sombor index, offering valuable insights into its behaviour and usage in both theoretical and practical graph analyses. This structured approach not only provides thorough proofs for the claims and theorems but also uses logical reasoning to eliminate other possibilities, ensuring the reduced Sombor index bounds are well established for various graph types. In future studies, attention can be directed toward trees showing either the minimal or maximal reduced Sombor index. Analyses may also investigate graphs with a secondary minimum or maximum. Furthermore, the constraints for trees based on the number of vertices and leaves could be assessed. 3. Methods The initial phase of our study establishes a universal bound for any graph, proving that the reduced Sombor index (𝑅𝑆𝑂) is always non-negative, with equality occurring only in specific cases like 𝐾2 or a disjoint union of 𝐾2 . For paths and stars with 𝑛 vertices, we calculate the 𝑅𝑆𝑂 by analyzing vertex degrees and structural properties. In paths, the linear structure simplifies the degree summation, while for stars, the central and leaf vertices have distinct degree contributions to the 𝑅𝑆𝑂. For wheel graphs, which consist of an outer cycle and a central hub vertex connected to all others, 𝑅𝑆𝑂 calculation is based on the interaction between the central hub and the outer cycle, by summing the degree contributions from both the hub and outer vertices. In graphs with cycles, the lower bound of 𝑅𝑆𝑂 is achieved when all vertices are on a single unique cycle, derived from analyzing vertex degree distributions and their cyclical nature. Additionally, we provide exact formulas for the 𝑅𝑆𝑂 in complete graphs and complete bipartite graphs, depending on the number of vertices and their partitions. For a complete graph with 𝑛 vertices, the Reduced Sombor index (𝑅𝑆𝑂) is given by 𝑅𝑆𝑂(𝐺) = 𝑛(𝑛 βˆ’ 1)(𝑛 βˆ’ 2)√2 . For a complete bipartite graph, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 125 https://internationalpubls.com the 𝑅𝑆𝑂 is calculated as 𝑅𝑆𝑂(𝐺) = π‘π‘žβˆš(𝑝 βˆ’ 1)2 + (π‘ž βˆ’ 1)2, where ∣ 𝑋 ∣= 𝑝 and ∣ π‘Œ ∣= π‘ž represent the two vertex partitions. These results are derived from analyzing the degree distributions in these regular structures, which facilitate the 𝑅𝑆𝑂 calculations. For connected graphs with weak support vertices, we demonstrate that 𝑅𝑆𝑂(𝐺) β‰₯ π‘š, where π‘š is the number of edges and π‘ž is the number of weak support vertices. This finding stems from analyzing how pendant vertices next to weak support vertices affect the graph's structure and 𝑅𝑆𝑂 value. Additionally, we establish a theorem identifying graphs with an 𝑅𝑆𝑂 of exactly zero through a case- by-case analysis of connected and disconnected graphs, utilizing vertex degree properties. We also prove that for graphs with 𝑛 β‰₯ 3 vertices, 𝑅𝑆𝑂(𝐺) β‰₯ 𝑛 βˆ’ 1, achieving equality only if the graph is a path or a star, based on graph diameters. To establish the results and bounds, we utilized several key graph theory techniques. The degree of each vertex is crucial for determining the Reduced Sombor index (𝑅𝑆𝑂). By analyzing vertex degrees across different graph structures, we derived precise bounds and values for the 𝑅𝑆𝑂. We conducted detailed case-by-case examinations of specific graph typesβ€”such as paths, cycles, trees, and stars to understand how their unique properties influence the 𝑅𝑆𝑂. Proof by contradiction helped us eliminate graph configurations that did not meet specific 𝑅𝑆𝑂 conditions, proving the uniqueness of certain structures for given bounds. Additionally, we applied logical reasoning regarding graph connectivity and diameter to identify necessary and sufficient conditions for specific 𝑅𝑆𝑂 values. These techniques allowed us to establish rigorous bounds on the 𝑅𝑆𝑂 for a wide variety of graphs. 4. Results Finding bounds of topological indices are always has been interesting and studied by researchers in the literature. For results on the bounds of various topological indices, the reader may refer [4]. For a wide collection of graphs, the bounds on the reduced Sombor index are observed in this section. These results are also used in the subsequent results where we found the maximum and minimum values of the reduced Sombor index of graphs. Let us start with the following proposition: Observation 1 For any graph 𝐺, 𝑅𝑆𝑂 β‰₯ 0. Proposition 2 If 𝐺 has a path on 𝑛 vertices, then, 𝑅𝑆𝑂(𝐺) = 𝑛 βˆ’ 1. Proposition 3 If 𝐺 is a star on 𝑛 vertices where 𝑛 β‰₯ 4, then, 𝑅𝑆𝑂(𝐺) = 𝑛 βˆ’ 1. Proposition 4 If 𝐺 is a wheel on 𝑛 vertices where 𝑛 β‰₯ 4, then, 𝑅𝑆𝑂(𝐺) = (𝑛 βˆ’ 1)(1 + √2) Proposition 5 If 𝐺 is a graph with at least one cycle, then, 𝑅𝑆𝑂(𝐺) β‰₯ 3√2. Proposition 6 If 𝐺 is a graph with a cycle of length π‘˜. Then, 𝑅𝑆𝑂(𝐺) β‰₯ π‘˜βˆš2 with equality if and only if every vertex of 𝐺 lies in the unique cycle of cycle. Proposition 7 If 𝐺 is a complete graph, then, 𝑅𝑆𝑂(𝐺) = 𝑛(𝑛 βˆ’ 1)(𝑛 βˆ’ 2)√2.. Proposition 8 If 𝐺 = (𝑋, π‘Œ ) is a complete bipartite graph with |𝑋| = 𝑝 and |π‘Œ | = π‘ž, then 𝑅𝑆𝑂(𝐺) = π‘π‘žβˆš(𝑝 βˆ’ 1)2 + (π‘ž βˆ’ 1)2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 126 https://internationalpubls.com Proposition 9 If 𝐺 is a connected graph with π‘ž weak support vertices, 𝑅𝑆𝑂(𝐺) β‰₯ (π‘š βˆ’ π‘ž), where π‘š is the number of edges in 𝐺. Proof: Let 𝐺 be a connected graph with π‘ž weak support vertices, say 𝑒1, 𝑒2, . . . , π‘’π‘ž. Then, there are π‘ž pendant vertices adjacent to these weak support vertices, say 𝑣1, 𝑣2, . . . , π‘£π‘ž. Then, out of the π‘š-edges of 𝐺, these π‘ž edges 𝑒1𝑣1, 𝑒2𝑣2, . . . π‘’π‘žπ‘£π‘ž are adding a zero for each, 3 and the remaining (π‘š βˆ’ π‘ž) edges add at least one for each implies that 𝑅𝑆𝑂(𝐺) = π‘š βˆ’ π‘ž. But, there may be edges (out of the (π‘š βˆ’ π‘ž) βˆ’ 𝑒𝑑𝑔𝑒𝑠) adding more than 1 also to 𝑅𝑆𝑂(𝐺), for example, if G have a consecutive 3 adjacent vertices 𝑒, 𝑣, 𝑀 such that 𝑑(𝑣) β‰₯ 3, then the edges 𝑒𝑣, 𝑣𝑀 both adds at least 2 to 𝑅𝑆𝑂(𝐺). Thus, 𝑅𝑆𝑂(𝐺) β‰₯ (π‘š βˆ’ π‘ž). β–‘ Theorem 10 If 𝐺 is a graph with the number of vertices 𝑛 β‰₯ 2, 𝑅𝑆𝑂(𝐺) β‰₯ 0 and the equality is true if and only if 𝐺 = 𝐾2 π‘œπ‘Ÿ 𝐺 = π‘šπΎ2. Proof: The converse of the result is true as 𝐺 = 𝐾2 π‘œπ‘Ÿ 𝐺 = π‘šπΎ2 are graphs with RSO(G) = 0. To prove the necessary part, first assume that RSO(G) = 0. Let π‘₯ be a vertex of 𝐺 such that 𝑑(π‘₯) β‰₯ 2. Let 𝑦 be a vertex incident with π‘₯ in 𝐺. Irrespective of the degree of 𝑦, the edge π‘₯𝑦 adds up a 1 to the 𝑅𝑆𝑂(𝐺), which implies that 𝑅𝑆𝑂(𝐺) β‰₯ 1, which is a contradiction to our assumption. Thus, 𝑑(π‘₯) cannot be more than one. Thus, any vertex of 𝐺 can have only degree one. Case-1: 𝐺 is connected. Claim: 𝐺 = 𝐾2. It is enough to prove that 𝐺 has only two vertices. Suppose that G let 𝑒, 𝑣, 𝑀 be 3 arbitrary vertices of 𝐺 such that 𝑒 and 𝑣 are adjacent and 𝑣 and 𝑀 are adjacent. Since 𝐺 is a connected graph, this assumption is possible, otherwise 𝐺 would be disconnected. Thus, 𝑑(𝑣) β‰₯ 2, implying that 𝑅𝑆𝑂 β‰₯ 2, a contradiction. Thus, 𝐺 has only two vertices, which means, 𝐺 = 𝐾2. Figure 1: Graphs G = 𝐾2 and G = m𝐾2 Case-2: 𝐺 is disconnected. Claim: 𝐺 = π‘šπΎ2. Since 𝐺 is disconnected, G must be the union of a finite number of connected components, say 𝐺1, 𝐺2, . . . , πΊπ‘š. For each of these connected component, by Case-1, we have that 𝐺𝑖 = 𝐾2.for 𝑖 = 1, 2, . . . , π‘š. Thus, 𝐺 = π‘šπΎ2. By Case-1 and Case-2, the theorem is proved. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 127 https://internationalpubls.com Theorem 11 If 𝐺 is a connected graph with the number of vertices 𝑛 β‰₯ 3, 𝑅𝑆𝑂(𝐺) β‰₯ 𝑛 βˆ’ 1 and the equality is true if and only if 𝐺 is a path on 𝑛 vertices, or 𝐺 is a star. Proof: Let 𝐺 be a graph with at least 3 vertices. If 𝐺 is a path on 𝑛 vertices, then by Proposition-2, 𝑅𝑆𝑂(𝐺) = 𝑛 βˆ’ 1, or if 𝐺 is a star on 𝑛 vertices, then by Proposition-3, 𝑅𝑆𝑂(𝐺) = 𝑛 βˆ’ 1. Thus, the sufficient part is true. To prove the other part, let 𝐺 be a graph with n vertices and 𝑅𝑆𝑂(𝐺) = 𝑛 βˆ’ 1. If 𝐺 is a cycle, then 𝑅𝑆𝑂(𝐺) β‰₯ 𝑛 √2 > 𝑛 βˆ’ 1. Thus, 𝐺 cannot even contain a cycle. Since 𝐺 is a connected graph, we must have that 𝐺 is a tree. We shall prove the result based on the diameter of 𝐺. Claim-1: If π‘‘π‘–π‘Žπ‘š(𝐺) = 2, then 𝐺 is a star. Assume that π‘‘π‘–π‘Žπ‘š(𝐺) = 2. Then, the path joining any two vertices is of length two. Since the graph is connected, there must be a vertex π‘₯ which is the center of all paths of length two. Otherwise, each pair of vertices are connected by a distinct path of length two which implies that 𝐺 is the union of copies of 𝑃3. Then, 𝐺 is disconnected, a contradiction. Figure 2: A star Thus, there exists a center vertex of all paths of length two. Now, if π‘₯1, π‘₯2, . . . , π‘₯π‘›βˆ’1 are the vertices connected to π‘₯, we claim that none of these vertices π‘₯𝑖π‘₯𝑗 , 1 ≀ 𝑖 ≀ (𝑛 βˆ’ 2), 𝑗 = 𝑖 + 1 are connected. Otherwise, it forms a wheel, and by Proposition-4, 𝑅𝑆𝑂(𝐺) β‰₯ (𝑛 βˆ’ 1), a contradiction. Thus, π‘₯ is the center vertex and all other π‘₯𝑖′s, 1 ≀ 𝑖 ≀ (𝑛 βˆ’ 1) are adjacent to π‘₯. This implies that 𝐺 is a star. Claim-2: If π‘‘π‘–π‘Žπ‘š(𝐺) β‰₯ 3, then 𝐺 is a path. Assume now that π‘‘π‘–π‘Žπ‘š(𝐺) β‰₯ 3. We claim that all vertices of 𝐺 belong to a unique diametrical path 𝑃 ∢ 𝑣1𝑣2, . . . , 𝑣𝑛 of 𝐺. Suppose there exists a vertex, say 𝑒 which is not on the diametrical path but adjacent to a vertex, say 𝑣, which is lying on the diametrical path. Without loss of generality, let 𝑣 be a support vertex of 𝐺. Assume that the diametrical path is given by 𝑣1𝑣2 = 𝑣, . . . π‘£π‘›βˆ’1. Thus, the vertex 𝑣2 = 𝑣 is the only vertex that has 𝑑(𝑣) = 3 and all other vertices have degree either one or two. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 128 https://internationalpubls.com For this structure, the reduced Sombor index is given by 𝑅𝑆𝑂(𝐺) = √(𝑑(𝑣1) βˆ’ 1)2 + (𝑑(𝑣2) βˆ’ 1)2 + √(𝑑(𝑣2) βˆ’ 1)2 + (𝑑(𝑒) βˆ’ 1)2 + √(𝑑(π‘£π‘›βˆ’2) βˆ’ 1)2 + (𝑑(π‘£π‘›βˆ’1) βˆ’ 1)2 + √(𝑑(𝑣2) βˆ’ 1)2 + (𝑑(𝑣3) βˆ’ 1)2 + βˆ‘ √(𝑑(𝑣𝑖) βˆ’ 1)2 + (𝑑(𝑣𝑖+1) βˆ’ 1)2 π‘›βˆ’2 𝑖=3 = √(1 βˆ’ 1)2 + (3 βˆ’ 1)2 + √(3 βˆ’ 1)2 + (1 βˆ’ 1)2 + √(2 βˆ’ 1)2 + (1 βˆ’ 1)2 + √(3 βˆ’ 1)2 + (2 βˆ’ 1)2 + √(2 βˆ’ 1)2 + (2 βˆ’ 1)2 = √22 + √22 + √12 + √5 + (𝑛 βˆ’ 4)√12 = 2 + 2 + 1 + √5 + (𝑛 βˆ’ 4) > 𝑛 – 1 The choice of 𝑣 being adjacent to any other vertex on the diametrical path results in the same 𝑅𝑆𝑂(𝐺), which implies that 𝑣 cannot be adjacent to a vertex on the diametrical path. So, there is no other vertex adjacent to a vertex 𝑀 which is adjacent to a vertex on the diametrical path. Thus, there cannot be two different diametrical paths in 𝐺. Thus, 𝐺 itself is a path. Thus, the theorem is proved. References [1] J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, 2008. [2] Haynes, T., Hedetniemi, S., Slater, P.:Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998). [3] Haynes T.W. Hedetniemi S. Slater P., Domination in Graphs: Advanced Topics, Marcel Dekker, 1998. [4] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021) 11–16. [5] H. Liu, I. Gutman, L. You, Y. Huang, Sombor index: review of extremal results and bounds, J. Math. Chem. 60 (2022) 771–798. [6] I. Gutman, Sombor index – one year later, Bull. Acad. Serb. Sci. Arts 153 (2020) 43–55. [7] H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022) 23–49. [8] X. Sun, J. Du, On Sombor index of trees with fixed domination number, Appl. Math. Comput. 421 (2022) #126946. [9] S. Li, Z. Wang, M. Zhang, On the extremal Sombor index of trees with a given diameter, Appl. Math. Comput. 416 (2022) #126731. [10] I. Gutman, V. R. Kulli, I. RedΛ‡zepoviΒ΄c, Sombor index of Kragujevac trees, Sci. Publ.Univ. Novi Pazar Ser. A 13 (2021) 61–70. [11] K. C. Das, I. Gutman, On Sombor index of trees, Appl. Math. Comput. 412 (2022)#126575. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 129 https://internationalpubls.com [12] I. RedΛ‡zepoviΒ΄c, Chemical applicability of Sombor indices, J. Serb. Chem. Soc. 86 (2021) 445–457. [13] H. Deng, Z. Tang, R. Wu, Molecular trees with extremal values of Sombor indices,Int. J. Quantum Chem. 121 (2021) #e26622. [14] I. Gutman, I. Redzepovic, B. Furtula, On the product of Sombor and modified Sombor index, Open Journal of Applied Discrete Mathematics, 6(2) (2023) 1-6. [15] H. Shoostari, S.M. Sheikholeslami, J. Amjadi, Modified Sombor index of unicyclic graphs with a given diameter, Asian-European Journal of Mathematics, 16(06) (2023) 2350098. [16] Yufei Huang, Hechao Liu, Bounds of modified Sombor index, spectral radius and energy, AIMS Mathematics, 6(10), (2021) 11263-11274. [17] Xuewe Zuo, Bilal Ahmed Rathar, Muhammad Imran, Akbar Ali, On some topological indices defined via the modified Sombor index, Molecules 27(19), (2022) 6772. [18] Fangxia Wang, Baoyindureng Wu, The Proof of a Conjecture on the Reduced Sombor Index, MATCH Commun. Math. Comput. Chem. 88 (2022) 583-591. [19] Hechao Liu, Lihua You, Zikai Tang, Jia–Bao Liu, On the Reduced Sombor Index and Its Applications, MATCH Commun. Math. Comput. Chem. 86 (2021) 729-753.