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Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1830
Author(s):  
Kinkar Chandra Das ◽  
Ali Ghalavand ◽  
Ali Reza Ashrafi

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.


2019 ◽  
Vol 11 (02) ◽  
pp. 1950025
Author(s):  
S. S. Bose ◽  
M. Nath ◽  
D. Sarma

In this paper, we determine the unique tree that maximizes the distance spectral radius in the class of all trees in which each non-pendent vertex has degree at least [Formula: see text].


2014 ◽  
Vol 06 (02) ◽  
pp. 1450028
Author(s):  
D. KALITA

The unicyclic graph Cn,gobtained by appending a cycle Cgof length g to a pendent vertex of a path on n - g vertices is the lollipop graph on n vertices. In [Algebraic connectivity of lollipop graphs, Linear Algebra Appl.434 (2011) 2204–2210], Guo et al. proved that a( Cn,g-1) < a( Cn,g) for g ≥ 4, where a( Cn,g) is the algebraic connectivity of Cn,g. In this paper, we present a new approach which is quite different from that of Guo et al. in proving a( Cn,g-1) < a( Cn,g) for g ≥ 4.


2013 ◽  
Vol 438 (11) ◽  
pp. 4260-4278 ◽  
Author(s):  
Surya Sekhar Bose ◽  
Milan Nath ◽  
Somnath Paul

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