Related papers: Theta-Relations Among Degree-Based Tree Indices
For a simple connected graph $G=(V,E)$, let $d(u)$ be the degree of the vertex $u$ of $G$. The general Sombor index of $G$ is defined as $$SO_{\alpha}(G)=\sum_{uv\in E} \left[d(u)^2+d(v)^2\right]^\alpha$$ where $SO(G)=SO_{0.5}(G)$ is the…
Motivated by the recently introduced topological index, the Somber index, we define a new topological index of a graph in this paper, we call it Sombor coindex. The Sombor coindex is defined by considering analogous contributions from the…
We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this…
In the study of topological indices two negative correlations are well known: that between the number of subtrees and the Wiener index (sum of distances), and that between the Merrifield-Simmons index (number of independent vertex subsets)…
A semiregular tree is a tree where all non-pendant vertices have the same degree. Belardo et al. (MATCH Commun. Math. Chem. 61(2), pp. 503-515, 2009) have shown that among all semiregular trees with a fixed order and degree, a graph with…
Topological indices play a significant role in mathematical chemistry. Given a graph $\mathcal{G}$ with vertex set $\mathcal{V}=\{1,2,\dots,n\}$ and edge set $\mathcal{E}$, let $d_i$ be the degree of node $i$. The degree-based topological…
We introduce the general Albertson irregularity index of a connected graph $G$ and define it as $A_{p}(G) =(\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}}$, where $p$ is a positive real number and $d(v)$ is the degree of the vertex $v$ in…
The zeroth-order general Randi\'c index (usually denoted by $R_{\alpha}^{0}$) and variable sum exdeg index (denoted by $SEI_{a}$) of a graph $G$ are defined as $R_{\alpha}^{0}(G)= \sum_{v\in V(G)} (d_{v})^{\alpha}$ and $SEI_{a}(G)=…
The Sombor index of a graph $G$, introduced by Ivan Gutman, is defined as the sum of the weights $\sqrt{d_G(u)^2+d_G(v)^2}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of vertex $u$ in $G$. The Sombor coindex is recently…
We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a…
Introduced by Gutman in 2021, the Sombor index is a novel graph-theoretic topological descriptor possessing potential applications in the modeling of thermodynamic properties of compounds. Let G^k_n be the set of all n-vertex connected…
Theta-graphs are a type of spatial graph with two vertices connected by three edges. We investigate embeddings of theta-graphs in the square and simple cubic lattices, using a combination of the Wang-Landau Monte Carlo method with a variant…
Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of…
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…
The atom-bond connectivity (ABC) index is a degree-based molecular descriptor with diverse chemical applications. Recent work of Lin et al. [W. Lin, J. Chen, C. Ma, Y. Zhang, J. Chen, D. Zhang, and F. Jia, On trees with minimal ABC index…
We propose a family of graph structural indices related to the Matrix-forest theorem. The properties of the basic index that expresses the mutual connectivity of two vertices are studied in detail. The derivative indices that measure…
The modified Albertson index, denoted by $A\!^*\!$, of a graph $G$ is defined as $A\!^*\!(G)=\sum_{uv\in E(G)} |(d_{u})^{2}- (d_{v})^{2}|$, where $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively, of $G$ and $E(G)$ is…
Topological indices are scientific details of graphs which represents its topology and of the most part graph invariant. In QSAR/QSPR, physico-chemical characteristics and topological indices, for example, atom bond connectivity (ABC) and…
The Sombor index is a topological index in graph theory defined by Gutman in 2021. In this article we find the maximum Sombor index of unicyclic graphs with a fixed number of pendant vertices. We also provide the unique graph among the…
We consider topological indices I that are sums of f(deg(u)) f(deg(v)), where {u,v} are adjacent vertices and f is a function. The Randi{\'c} connectivity index or the Zagreb group index are examples for indices of this kind. In earlier…