Research Article Sharp Lower Bounds of the Sum-Connectivity Index of Unicyclic Graphs Maryam Atapour Department of Mathematics and Computer Science, Basic Science Faculty, University of Bonab, P.O. Box 55513-95133, Bonab, Iran Correspondence should be addressed to Maryam Atapour; m.atapour@ubonab.ac.ir Received 1 August 2021; Revised 17 August 2021; Accepted 23 August 2021; Published 2 September 2021 Academic Editor: Ali Ahmad Copyright © 2021 Maryam Atapour. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e sum-connectivity index of a graph G is defined as the sum of weights 1/ ������ d u + d v  over all edges uv of G, where d u and d v are the degrees of the vertices u and v in graph G, respectively. In this paper, we give a sharp lower bound on the sum-connectivity index unicyclic graphs of order n ≥ 7 and diameter D(G) ≥ 5. 1. Introduction and Preliminaries Let G beasimplegraphwithavertexset V � V(G) and edge set E(G). e integers n � n(G)�|V(G)| and m � m(G)� |E(G)| aretheorderandthesizeofthegraph G, respectively. e open neighborhood of vertex v is N G (v)� N(v)� u ∈ V(G)|uv ∈ E(G) { }, and the degree of v is d G (v)� d v � |N(v)|. A pendant vertex is a vertex of degree one. e distance between two vertices is the number of edges in the shortest path connecting them, and the diameter D(G) of G is the distance between any two furthest vertices in G.A diametral path is the shortest path in G connecting two vertices whose distance is D(G). A unicyclic graph is a connected graph containing exactly one cycle. A subgraph G ′ of a graph G is a graph whose set of vertices is a subset of V(G), and set of edges is a subset of E(G). Atopologicalindexisanumericnumberassociatedwith a molecular graph that correlates certain physicochemical properties of chemical compounds. e topological indices are useful in the prediction of physicochemical properties and the bioactivity of the chemical compounds [1–3]. Also, topological indices invariants are used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) studies. It was demonstrated that the sum-connectivity index is well cor- related with a variety of physicochemical properties of alkanes,suchasboilingpointandenthalpyofformation.e sum-connectivity index is certainly the most widely applied in chemistry and pharmacology, in particular for designing quantitative structure-property and structure-activity rela- tions. e sum-connectivity index is proposed to quanti- tatively characterize the degree of molecular branching. Topologicalindiceshavebeenusedandhavebeenshown to give a high degree of predictability of pharmaceutical properties. e sum-connectivity index of a graph G was proposed in [4] defined as follows: SCI(G)�  uv∈E(G) 1 ������ d u + d v  . (1) e applications of the sum-connectivity index have been investigated in [5, 6]. Some basic mathematical properties of the sum-connectivity index have been estab- lished in [4–8]. In[4],itwasshownthatforagraph G with n ≥ 5 vertices and without isolated vertices, SCI(G) ≥ n − 1/ � n √ with equality if and only if G is the star. For n � 4, this is not true since,fortheunionoftwocopiesofthepathontwovertices, its sum-connectivity index is � 2 √ , less than 3/2. In [7], minimum sum-connectivity indices of trees and unicyclic graphsofagivenmatchingnumberarecharacterized;in[8], sum-connectivity index of molecular trees are characterized; Hindawi Journal of Mathematics Volume 2021, Article ID 8391480, 6 pages https://doi.org/10.1155/2021/8391480