On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices
The general sum-connectivity index of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }(G)$ </tex-math></inline-formula>, i...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
IEEE
2019-01-01
|
| Series: | IEEE Access |
| Subjects: | |
| Online Access: | https://ieeexplore.ieee.org/document/8823929/ |
| _version_ | 1811212269908918272 |
|---|---|
| author | Akbar Ali Shahzad Ahmed Zhibin Du Wei Gao Muhammad Aslam Malik |
| author_facet | Akbar Ali Shahzad Ahmed Zhibin Du Wei Gao Muhammad Aslam Malik |
| author_sort | Akbar Ali |
| collection | DOAJ |
| description | The general sum-connectivity index of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }(G)$ </tex-math></inline-formula>, is defined as <inline-formula> <tex-math notation="LaTeX">$\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$uv$ </tex-math></inline-formula> is the edge connecting the vertices <inline-formula> <tex-math notation="LaTeX">$u,v\in V(G)$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$d(w)$ </tex-math></inline-formula> denotes the degree of a vertex <inline-formula> <tex-math notation="LaTeX">$w\in V(G)$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> is a non-zero real number. For <inline-formula> <tex-math notation="LaTeX">$\alpha =-1/2$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n\geq 11$ </tex-math></inline-formula>, Wang <italic>et al.</italic> [On the sum-connectivity index, Filomat 25 (2011) 29–42] proved that <inline-formula> <tex-math notation="LaTeX">$K_{2} + \overline {K}_{n-2}$ </tex-math></inline-formula> is the unique graph with minimum <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }$ </tex-math></inline-formula> value among all the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>–vertex graphs having minimum degree at least 2, where <inline-formula> <tex-math notation="LaTeX">$K_{2} + \overline {K}_{n-2}$ </tex-math></inline-formula> is the join of the 2-vertex complete graph <inline-formula> <tex-math notation="LaTeX">$K_{2}$ </tex-math></inline-formula> and the edgeless graph <inline-formula> <tex-math notation="LaTeX">$\overline {K}_{n-2}$ </tex-math></inline-formula> on <inline-formula> <tex-math notation="LaTeX">$n-2$ </tex-math></inline-formula> vertices. Tomescu [2-connected graphs with minimum general sum-connectivity index, Discrete Appl. Math. 178 (2014) 135–141] proved that the result of Wang <italic>et al.</italic> holds also for <inline-formula> <tex-math notation="LaTeX">$n\geq 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$-1\leq \alpha < -0.867$ </tex-math></inline-formula>. In this paper, it is shown that the aforementioned result of Wang <italic>et al.</italic> remains valid if the graphs under consideration are connected, <inline-formula> <tex-math notation="LaTeX">$n\geq 6$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$-1\leq \alpha < \alpha _{0}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\alpha _{0}\approx -0.68119$ </tex-math></inline-formula> is the unique real root of the equation <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }(K_{2} + \overline {K}_{4}) - \chi _{_\alpha }(C_{6})=0$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$C_{6}$ </tex-math></inline-formula> is the cycle on 6 vertices. |
| first_indexed | 2024-04-12T05:26:23Z |
| format | Article |
| id | doaj.art-0ad05ccbdd914853b7c53339a20b41f9 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-04-12T05:26:23Z |
| publishDate | 2019-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-0ad05ccbdd914853b7c53339a20b41f92022-12-22T03:46:15ZengIEEEIEEE Access2169-35362019-01-01713674313675110.1109/ACCESS.2019.29395108823929On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant VerticesAkbar Ali0https://orcid.org/0000-0001-8160-4196Shahzad Ahmed1Zhibin Du2https://orcid.org/0000-0001-5795-3580Wei Gao3https://orcid.org/0000-0001-7963-3502Muhammad Aslam Malik4Knowledge Unit of Science, University of Management and Technology, Sialkot, PakistanKnowledge Unit of Science, University of Management and Technology, Sialkot, PakistanSchool of Mathematics and Statistics, Zhaoqing University, Zhaoqing, ChinaSchool of Information Science and Technology, Yunnan Normal University, Kunming, ChinaDepartment of Mathematics, University of the Punjab, Quaid-i-Azam Campus, Lahore, PakistanThe general sum-connectivity index of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }(G)$ </tex-math></inline-formula>, is defined as <inline-formula> <tex-math notation="LaTeX">$\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$uv$ </tex-math></inline-formula> is the edge connecting the vertices <inline-formula> <tex-math notation="LaTeX">$u,v\in V(G)$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$d(w)$ </tex-math></inline-formula> denotes the degree of a vertex <inline-formula> <tex-math notation="LaTeX">$w\in V(G)$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> is a non-zero real number. For <inline-formula> <tex-math notation="LaTeX">$\alpha =-1/2$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n\geq 11$ </tex-math></inline-formula>, Wang <italic>et al.</italic> [On the sum-connectivity index, Filomat 25 (2011) 29–42] proved that <inline-formula> <tex-math notation="LaTeX">$K_{2} + \overline {K}_{n-2}$ </tex-math></inline-formula> is the unique graph with minimum <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }$ </tex-math></inline-formula> value among all the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>–vertex graphs having minimum degree at least 2, where <inline-formula> <tex-math notation="LaTeX">$K_{2} + \overline {K}_{n-2}$ </tex-math></inline-formula> is the join of the 2-vertex complete graph <inline-formula> <tex-math notation="LaTeX">$K_{2}$ </tex-math></inline-formula> and the edgeless graph <inline-formula> <tex-math notation="LaTeX">$\overline {K}_{n-2}$ </tex-math></inline-formula> on <inline-formula> <tex-math notation="LaTeX">$n-2$ </tex-math></inline-formula> vertices. Tomescu [2-connected graphs with minimum general sum-connectivity index, Discrete Appl. Math. 178 (2014) 135–141] proved that the result of Wang <italic>et al.</italic> holds also for <inline-formula> <tex-math notation="LaTeX">$n\geq 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$-1\leq \alpha < -0.867$ </tex-math></inline-formula>. In this paper, it is shown that the aforementioned result of Wang <italic>et al.</italic> remains valid if the graphs under consideration are connected, <inline-formula> <tex-math notation="LaTeX">$n\geq 6$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$-1\leq \alpha < \alpha _{0}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\alpha _{0}\approx -0.68119$ </tex-math></inline-formula> is the unique real root of the equation <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }(K_{2} + \overline {K}_{4}) - \chi _{_\alpha }(C_{6})=0$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$C_{6}$ </tex-math></inline-formula> is the cycle on 6 vertices.https://ieeexplore.ieee.org/document/8823929/Chemical graph theorygeneral sum-connectivity indextopological index |
| spellingShingle | Akbar Ali Shahzad Ahmed Zhibin Du Wei Gao Muhammad Aslam Malik On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices IEEE Access Chemical graph theory general sum-connectivity index topological index |
| title | On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices |
| title_full | On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices |
| title_fullStr | On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices |
| title_full_unstemmed | On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices |
| title_short | On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices |
| title_sort | on the minimal general sum connectivity index of connected graphs without pendant vertices |
| topic | Chemical graph theory general sum-connectivity index topological index |
| url | https://ieeexplore.ieee.org/document/8823929/ |
| work_keys_str_mv | AT akbarali ontheminimalgeneralsumconnectivityindexofconnectedgraphswithoutpendantvertices AT shahzadahmed ontheminimalgeneralsumconnectivityindexofconnectedgraphswithoutpendantvertices AT zhibindu ontheminimalgeneralsumconnectivityindexofconnectedgraphswithoutpendantvertices AT weigao ontheminimalgeneralsumconnectivityindexofconnectedgraphswithoutpendantvertices AT muhammadaslammalik ontheminimalgeneralsumconnectivityindexofconnectedgraphswithoutpendantvertices |