‎The Sombor index is a newly introduced vertex-degree-based graph invariant with the ability to predict the enthalpy‎‎of vaporization and entropy of octane isomers‎. ‎Recently‎, ‎two new variants of the Sombor index namely the reduced and increased Sombor indices were put forward‎. ‎The reduced and increased Sombor indices are respectively defined for graph $\Gamma$ as‎$$SO_{red}(\Gamma)=\sum_{\mathcal{FG}\inE(\Gamma)}\sqrt{(d_{\Gamma}(\mathcal{F})-1)^2+(d_{\Gamma}(\mathcal{G})-1)^2},$$‎ and$$SO^{\ddagger}(\Gamma)=\sum_{\mathcal{FG}\inE({\Gamma})}\sqrt{(d_{\Gamma}(\mathcal{F})+1)^2+(d_{\Gamma}(\mathcal{G})+1)^2},$$‎‎ in which $d_{\Gamma}(\mathcal{F})$ is the degree of the vertex $\mathcal{F}$ in $\Gamma$‎.‎ Our purpose is to establish sharp lower bounds on the reduced and increased Sombor indices of trees in terms of their order and maximum vertex degree‎. Moreover‎, ‎the extremal trees that attain the bounds are characterized‎.

For a graph G, the third neighborhood degree index of G is defined as: ND3(G) = X ab∈E(G) δG(a)δG(b)  δG(a) + δG(b) , where δG(a) represents the sum of degrees of all neighboring vertices of vertex a. In this short paper, we establish a new lower bound on the third neighborhood degree index of trees and characterize the extremal trees achieving this bound.

‎Bucket recursive trees are an interesting and natural generalization of recursive trees‎. ‎In this model the nodes are buckets that can hold up to b≥ 1 labels‎. ‎The (modified) Zagreb index of a graph is defined as the sum of‎ ‎the squares of the outdegrees of all vertices in the graph‎. ‎We give the mean and variance of this index in random bucket recursive trees‎. ‎Also‎, ‎two limiting results on this index are given‎.

A hypertree is a special type of connected hypergraph that removes‎ ‎any‎, ‎its hyperedge then results in a disconnected hypergraph‎. ‎Relation between hypertrees (hypergraphs) and trees (graphs) can be helpful to solve real problems in hypernetworks and networks and it is the main tool in this regard‎. ‎The purpose of this paper is to introduce a positive relation (as $\alpha$-relation) on hypertrees that makes a connection between hypertrees and trees‎. ‎This relation is dependent on some parameters such as path‎, ‎length of a path‎, ‎and the intersection of hyperedges‎. ‎For any $q\in \mathbb{N}‎, ‎$ we introduce the concepts of a derivable tree‎, ‎$(\alpha‎, ‎q)$-hypergraph‎, ‎and fundamental $(\alpha‎, ‎q)$-hypertree for the first time in this study and analyze the structures of derivable trees from hypertrees via given positive relation‎. ‎In the final‎, ‎we apply the notions of derivable trees‎, ‎$(\alpha‎, ‎q)$-trees in real optimization problems by modeling hypernetworks and networks based on hypertrees and trees‎, ‎respectively.‎‎‎

In this paper‎, ‎the characteristic polynomial and the spectrum of the terminal distance matrix for some Kragujevac trees is computed. ‎As Application‎, ‎we obtain an upper bound and a lower bound for the spectral radius of the terminal distance matrix of the Kragujevac trees‎.

‎Cruz‎, ‎Monsalve and Rada [Extremal values of vertex-degree-based topological indices of chemical trees‎, ‎Appl‎. ‎Math‎. ‎Comput‎. ‎380 (2020) 125281] posed an open problem to find the maximum value of the exponential second Zagreb index for chemical trees of given order‎. ‎In this paper‎, ‎we solve the open problem completely‎.

In this article‎, ‎we discuss the vertex decomposability of three well-studied simplicial complexes associated to forests‎. ‎In particular‎, ‎we show that the bounded degree complex of a forest and the complex of directed trees of a multidiforest is vertex decomposable‎. ‎We then prove that the non-cover complex of a forest is either contractible or homotopy equivalent to a sphere‎. ‎Finally we provide a complete characterization of forests whose non-cover complexes are vertex decomposable‎.

‎The additively weighted edge Mostar index is a topological index(TI) defined as an extension of the edge Mostar index‎. ‎In this paper‎, ‎we determine the extrema of the additively weighted edge Mostar index for trees‎. ‎Additionally‎, ‎we compute the lower bound and first four upper bounds of additively weighted edge Mostar index of unicyclic graphs and the upper bound for cacti with a fixed number of cycles‎. ‎All the graphs attaining the bounds are characterized‎. ‎We also propose two conjectures on additively weighted edge Mostar index of bicyclic graphs‎.

‎The geometric-quadratic index (GQ) was defined in 2021 by V‎. ‎R‎. ‎Kulli‎. ‎In a recent study‎, ‎QSPR analysis for the octane isomers of GQ and some other newly defined topological indices was presented‎. ‎This analysis has revealed that GQ dominates over many of the well-known topological indices in terms of chemical applicability potential‎, ‎especially for the heat of vaporization‎. ‎These results inspired us to investigate the mathematical properties of GQ‎. ‎In this paper‎, ‎extremal graphs for GQ are investigated among connected graphs‎, ‎trees‎, ‎and unicyclic graphs‎. ‎In addition‎, ‎several mathematical relations between GQ and some well-known topological indices are presented.

Assume denotes a connected and simple graph with edge set  E(G) as well as vertex set ‎V(G). ‎In chemical graph theory‎, ‎the atom-bond connectivity index, as well as the Randić index of graph are two well-defined topological indices‎. ‎In addition‎, ‎Ali and Du [On the difference between  ABC and Randić indices of binary and chemical trees‎, ‎Int‎. ‎J‎. ‎Quantum Chem‎. ‎(2017) e25446] recently unveiled the distinction between Randić and ABC indices‎. ‎In this report‎, ‎we study the link between the difference of Randić and ABC indices with certain well-studied topological indices‎.