Minimum sum-connectivity indices of trees and unicyclic graphs of a given matching number

Abstract

We obtain the maximum sum-connectivity indices of graphs in the set of trees and in the set of unicyclic graphs respectively with given number of vertices and maximum degree, and determine the corresponding extremal graphs. Additionally, we deduce the n-vertex unicyclic graphs with the first two maximum sum-connectivity indices for 4. n 

Degree-Based Topological Indices

“…A number of properties of the sum-connectivity index have been determined, 125,[154][155][156][157][158][159] which again are bounds and characterization of graphs of various types, extremal with respect to SCI; details are found in the review. 160 By comparing the product-and sumconnectivity indices, 10,11,161,162 it was found that these have remarkably similar correlation properties.…”

Relations Between the Product- and Sum-connectivity Indices

“…Mathematical properties of the sum-connectivity index have also been established. [18][19][20][21] Related to the sum-connectivity index, sum-connectivity matrix and sum-connectivity energy have been proposed. 22 In this note, we establish several relations between the product-and the sum-connectivity indices.…”

Some graphs determined by their (signless) Laplacian spectra

“…S(n, c, k) have been investigated in many papers. For instance, S(n, c, 0) is the unique graph with the maximal spectral radius [1] (or the Merrifield-Simmons index [19]), the minimal Hosoya index (or the Wiener index [19], the Randić index [19]) in the set of all connected cacti on n vertices with c cycles, and S(n, 0, β − 1) is the unique tree with the maximum Laplacian Estrada index [8], and the minimum Laplacian-energy-like invariant [15], (or the Wiener index [9], the hyper-Wiener index [28]) in the class of trees with n vertices and the matching number β, where 2 β ⌊ 1 2 n⌋. Moreover, S(n, 1 2 (n − k) − 1, 1) is also an extremal graph [17] with the maximum signless Laplacian spectral radius in the class of connected cacti with n vertices and k pendant vertices.…”