Extremal values of the Sombor index in unicyclic and bicyclic graphs

Cruz, Roberto Moraes; Rada, Juan

doi:10.1007/s10910-021-01232-8

Cited by 50 publications

(24 citation statements)

References 14 publications

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“…The functions h 1 (x) = 10) and (11), we obtain SO(G) − SO(G ) < 0 and SO red (G) − SO red (G ) < 0, contradicted by the fact that SO(G) (resp. SO red (G)) is maximum.…”

Section: The Case Of ν =mentioning

confidence: 72%

“…Cruz et al [10] characterized the extremal graphs with respect to the Sombor index over the set of all n-vertex chemical graphs, connected chemical graphs, chemical trees, and hexagonal systems. Cruz and Rada [11], studied the extremal values of Sombor index over the set of all unicyclic and also bicyclic graphs of a given order. In a recent work, Das et al [12] obtained lower and upper bounds for the Sombor index of graphs based on some other graph parameters.…”

Section: Background Materialsmentioning

confidence: 99%

See 1 more Smart Citation

On a Conjecture about the Sombor Index of Graphs

2021

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Let G be a graph with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 and SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) is the degree of the vertex v in G. We denote by Hn,ν the graph constructed from the star Sn by adding ν edge(s), 0≤ν≤n−2, between a fixed pendent vertex and ν other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph Hn,ν has the maximum Sombor index among all connected ν-cyclic graphs of order n, where 0≤ν≤n−2. In some earlier works, the validity of this conjecture was proved for ν≤5. In this paper, we confirm that this conjecture is true, when ν=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n−ν−2 is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained.

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“…The functions h 1 (x) = 10) and (11), we obtain SO(G) − SO(G ) < 0 and SO red (G) − SO red (G ) < 0, contradicted by the fact that SO(G) (resp. SO red (G)) is maximum.…”

Section: The Case Of ν =mentioning

confidence: 72%

Section: Background Materialsmentioning

confidence: 99%

On a Conjecture about the Sombor Index of Graphs

2021

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“…Because this graph invariant is based on using Euclidean metrics, it promptly attracted the attention of quite a few colleagues. As a consequence of this, numerous papers on Sombor index have already been published (Alikhani & Ghanbari, 2021), (Cruz & Rada, 2021), (Došlić et al, 2021), (Horoldagva & Xu, 2021), (Kulli, 2021), and more will appear in the near future. Bearing this in mind, we were motivated to investigate the matrix constructed from the Sombor index in an earlier proposed manner (Das et al, 2018), and to study some of its spectral properties.…”

Section: Spectrum and Energy Of The Sombor Matrix Introductionmentioning

confidence: 97%

Spectrum and energy of the Sombor matrix

Gutman¹

2021

Vojnotehnički glasnik

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Introduction/purpose: The Sombor matrix is a vertex-degree-based matrix associated with the Sombor index. The paper is concerned with the spectral properties of the Sombor matrix. Results: Equalities and inequalities for the eigenvalues of the Sombor matrix are obtained, from which two fundamental bounds for the Sombor energy (= energy of the Sombor matrix) are established. These bounds depend on the Sombor index and on the "forgotten" topological index. Conclusion: The results of the paper contribute to the spectral theory of the Sombor matrix, as well as to the general spectral theory of matrices associated with vertex-degree-based graph invariants.

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“…In ref. [6], the extremal values of the Sombor index in unicyclic and bicyclic graphs were studied. Das, Cevik, Cangul and Shang presented lower and upper bounds on the Sombor index of graphs by using some graph parameters and obtain several relations on Sombor index with the first and second Zagreb indices of graphs [7].…”

Section: Introductionmentioning

confidence: 99%

The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs

Fang

You

Liu

2021

Int J of Quantum Chemistry

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Hexagonal chains are a special class of catacondensed benzenoid system and phenylene chains are a class of polycyclic aromatic compounds. Recently, A family of Sombor indices was introduced by Gutman in the chemical graph theory. It had been examined that these indices may be successfully applied on modeling thermodynamic properties of compounds. In this paper, we study the expected values of the Sombor indices in random hexagonal chains, phenylene chains, and consider the Sombor indices of some chemical graphs such as graphene, coronoid systems and carbon nanocones.

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Extremal values of the Sombor index in unicyclic and bicyclic graphs

Cited by 50 publications

References 14 publications

On a Conjecture about the Sombor Index of Graphs

On a Conjecture about the Sombor Index of Graphs

Spectrum and energy of the Sombor matrix

The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs

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