On a conjecture regarding the exponential reduced Sombor index of chemical trees

Discrete Math. Lett. 9 (2022) 107–110 DOI: 10.47443/dml.2021.s217 Discrete Mathematics Letters www.dmlett.com Research Article On a conjecture regarding the exponential reduced Sombor index of chemical trees∗ Amjad E. Hamza, Akbar Ali† Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il, Saudi Arabia (Received: 30 April 2022. Received in revised form: 5 May 2022. Accepted: 15 May 2022. Published online: 19 May 2022.) c 2022 the authors. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract √ 2 2 Let G be a graph and denote by du the degree of a vertex u of G. The sum of the numbers e (du −1) +(dv −1) over all edges uv of G is known as the exponential reduced Sombor index. A chemical tree is a tree with the maximum degree at most 4. In this paper, a conjecture posed by Liu et al. [MATCH Commun. Math. Comput. Chem. 86 (2021) 729–753] is disproved and its corrected version is proved. Keywords: topological index; chemical graph theory; Sombor index; reduced Sombor index; exponential reduced Sombor index. 2020 Mathematics Subject Classification: 05C05, 05C07, 05C92. 1. Introduction Let G be a graph. The sets of edges and vertices of G are represented by E(G) and V (G), respectively. For the vertex v ∈ V (G), the degree of v is denoted by dG (v) (or simply by dv if only one graph is under consideration). A vertex u ∈ V (G) is said to be a pendent vertex if du = 1. The degree set of G is the set of all unequal degrees of vertices of G. The set NG (u) consists of the vertices of the graph G that are adjacent to the vertex v. The members of NG (u) are known as neighbors of u. A chemical tree is the tree of maximum degree at most 4. The (chemical-)graph-theoretical terminology and notation that are used in this study without explaining here can be found in the books [1, 2, 11]. For the graph G, the Sombor index and reduced Sombor index abbreviated as SO and SOred , respectively, are defined [5] as X p X p SO(G) = d2u + d2v and SOred (G) = (du − 1)2 + (dv − 1)2 . uv∈E(G) uv∈E(G) These degree-based graph invariants, introduced recently in [5], have attained a lot of attention from researchers in a very short time, which resulted in many publications; for example, see the review papers [4, 9], and the papers listed therein. The following exponential version of the reduced Sombor index was considered in [10]: √ X 2 2 eSOred (G) = e (du −1) +(dv −1) . uv∈E(G) Let ni denote the number of vertices in the graph G with degree i. The cardinality of the set consisting of the edges joining the vertices of degrees i and j in the graph G is denoted by mi,j . Denote by Tn the class of chemical trees of order n such that n2 + n3 ≤ 1 and m1,3 = m1,2 = 0. Deng et al. [3] proved that the members of the class Tn are the only trees possessing the maximum value of the reduced Sombor index for every n ≥ 11. Keeping in mind this result of Deng et al. [3], Liu et al. [10] posed the following conjecture concerning the exponential reduced Sombor index for chemical trees. Conjecture 1.1. [10] Among all chemical trees of a fixed order n, the members of the class Tn are the only trees possessing the maximum value of the exponential reduced Sombor index for every n ≥ 11. Conjecture 1.1 was also discussed in [12] and was left open. In fact, there exist counter examples to Conjecture 1.1; for instance, for the trees T1 and T2 depicted in Figure 1, it holds that √ 278 ≈ eSOred (T1 ) = 8e3 + e3 2 √ + 2e 10 √ < e + 7e3 + 2e3 2 √ +e 10 = eSOred (T2 ) ≈ 306. The next theorem gives a corrected statement of Conjecture 1.1. ∗ This paper is dedicated to the memory of Professor Nenad Trinajstić (one of the pioneers of chemical graph theory). author (). † Corresponding A. E. Hamza and A. Ali / Discrete Math. Lett. 9 (2022) 107–110 108 T1 T2 Figure 1: The trees T1 and T2 providing a counterexample to Conjecture 1.1. Theorem 1.1. For n ≥ 7, if T is a chemical tree of order n, then   √  √ 1 10 3 3 2   + 3e if n ≡ 0 (mod 3) 3e − 5e − e   3     √  √  1 3 1 3 √ √  eSOred (T ) ≤ 2e + e3 2 n + 2e − 5e3 2 + 1 6e2 − 7e3 − 2e3 2 + 3e 13 if n ≡ 1 (mod 3)  3 3  3     0 if n ≡ 2 (mod 3), with equality if and only if • the degree set of T is {1, 2, 4} and n2 = m2,4 = m1,2 = 1, whenever n ≡ 0 (mod 3); • the degree set of T is {1, 3, 4} and n3 = m3,4 = 1 and m1,3 = 2, whenever n ≡ 1 (mod 3); • the degree set of T is {1, 4} whenever n ≡ 2 (mod 3). 2. Proof of Theorem 1.1 If T is a chemical tree of order n with n ≥ 3, then eSOred (T ) = X √ mi,j e (i−1)2 +(j−1)2 , (1) 1≤i≤j≤4 n1 + n2 + n3 + n4 = n , (2) n1 + 2n2 + 3n3 + 4n4 = 2(n − 1) , (3) mj,i + 2mj,j = j · nj (4) X for j = 1, 2, 3, 4. 1≤i≤4 i6=j By solving the system of equations (2)–(4) for the unknowns m1,4 , m4,4 , n1 , n2 , n3 , n4 and then inserting the values of m4,4 and m1,4 (these two values are well-known, see for example [6]) in Equation (1), one gets eSOred (T ) = √  √  √  1 3 1 1 3 2e + e3 2 n + 2e − 5e3 2 + 3e − 4e3 + e3 2 m1,2 3 3 3    √ √ √  1 1 + 9e2 − 10e3 + e3 2 m1,3 + 3e 2 − 2e3 − e3 2 m2,2 9 3   √ √  √ 1  √10 1 9e 5 − 4e3 − 5e3 2 m2,3 + 3e − e3 − 2e3 2 m2,4 + 9 3   √ √ √  1 1  √13 + 9e2 2 − 2e3 − 7e3 2 m3,3 + 9e − e3 − 8e3 2 m3,4 . 9 9 (5) We take √  1 3e − 4e3 + e3 2 m1,2 3 √  √ √  1 2 1 + 9e − 10e3 + e3 2 m1,3 + −2e3 + 3e 2 − e3 2 m2,2 9 3   √ √ √ √  1 1 3 + −4e3 − 5e3 2 + 9e 5 m2,3 + −e − 2e3 2 + 3e 10 m2,4 9 3   √ √ √ √  1 1 3 + −2e3 + 9e2 2 − 7e3 2 m3,3 + −e − 8e3 2 + 9e 13 m3,4 . 9 9 ≈ −0.8653m1,2 − 7.1958m1,3 − 32.4742m2,2 − 38.2323m2,3 Γ(T ) = − 29.4651m2,4 − 41.6713m3,3 − 27.2888m3,4 . (6) (7) A. E. Hamza and A. Ali / Discrete Math. Lett. 9 (2022) 107–110 109 Then, Equation (5) can be written as eSOred (T ) = √  √  1 3 1 3 2e + e3 2 n + 2e − 5e3 2 + Γ(T ) . 3 3 (8) For any given integer n greater than 4, it is evident from Equation (8) that a tree T attains the greatest value of eSOred over the class of all chemical trees of order n if and only if T possess the greatest value of Γ in the considered class. As a consequence, we consider Γ(T ) instead of eSOred (T ) in the next lemma. Lemma 2.1. Let T be a chemical tree of order n, where n ≥ 7. The inequality Γ(T ) < √ √  1 2 6e − 7e3 − 2e3 2 + 3e 13 (≈ −41.6804), 3 holds if any of the following conditions holds: (i) max{m3,3 , m2,2 , m2,3 } ≥ 1, (ii) max{m3,4 , m2,4 } ≥ 2, (iii) n2 + n3 ≥ 2. Proof. Take an edge uv ∈ E(T ) with du , dv ∈ {2, 3}. Since n ≥ 7, at least one of the two vertices u, v has at least two nonpendent neighbors. Hence, if max{m3,3 , m2,2 , m2,3 } ≥ 1 then either m3,3 + m2,2 + m2,3 ≥ 2 or max{m3,4 , m2,4 } ≥ 1 and hence the required inequality follows from (6). Also, note that the desired inequality follows from (6) whenever max{m3,4 , m2,4 } ≥ 2. In what follows, assume that m3,3 = m2,2 = m2,3 = 0, n2 + n3 ≥ 2, and max{m3,4 , m2,4 } ≤ 1. Assume that n3 6= 0. Let w ∈ V (...truncated)