Topological indices are a class of numerical invariants that predict certain physical and chemical properties of molecules. Recently, two novel topological indices, named as Sombor index and reduced Sombor index, were introduced by Gutman, defined as $$SO(G)=\sum_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)},$$ $$SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(d_{G}(u)-1)^{2}+(d_{G}(v)-1)^{2}},$$ where $d_{G}(u)$ denotes the degree of vertex $u$ in $G$.
In this paper, our aim is to order the chemical trees, chemical unicyclic graphs, chemical bicyclic graphs and chemical tricyclic graphs with respect to Sombor index and reduced Sombor index. We determine the first fourteen minimum chemical trees, the first four minimum chemical unicyclic graphs, the first three minimum chemical bicyclic graphs, the first seven minimum chemical tricyclic graphs. At last, we consider the applications of reduced Sombor index to octane isomers.
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.
The Sombor indices, a new category of degree-based topological molecular descriptors, have been widely investigated due to their excellent chemical applicability. This paper aims to establish Sombor indices distributions in random polygonal chain networks and to achieve expressions of the expected values and variances. The expected values and variances of the Sombor indices for polyonino, pentachain, polyphenyl, and cyclooctane chains are obtained. Since the end connection of a random chain network follows a binomial distribution, the Sombor indices of any chain network follow the normal distr…
We perform a detailed computational study of the recently introduced Sombor indices on random graphs. Specifically, we apply Sombor indices on three models of random graphs: Erdös-Rényi graphs, random geometric graphs, and bipartite random graphs. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the graph, scale with the graph average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random graphs and, as a consequence, we show that selected normalized Sombor indices …
In this work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically, we have inequalities for these indices relating them with other indices: the first Zagreb index, the forgotten index and the first variable Zagreb index. Finally, we solve some extremal problems for general Sombor indices.
Sombor index is a novel topological index, which was introduced by Gutman and defined for a graph $G$ as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}=d_{G}(u)$ denotes the degree of vertex $u$ in graph $G$. Extremal problems on the Sombor index for trees with a given diameter has been considered by Chen et al. [H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022) 23--49] and Li et al. [S. Li, Z. Wang, M. Zhang, On the extremal Sombor index of trees with a given diameter, Appl. Math. Comput. 416 (2022) 126…
We introduce a degree-based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: $mSO_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α+d_v^α\right) /2 \right]^{1/α}$. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0\}$. We also consider the limit cases $mSO_{α\to 0}(G)$ and $mSO_{α\to\pm\infty}(G)$. Indeed, for given values of $α$, the mean Sombor index is related to well-known topological indices such as the inverse sum ind…
We present the bounds in terms of other important graph parameters for general Sombor index which generalises both the forgotten index and the Sombor index. We also explore the Nordhaus-Gaddum-type result for the general Sombor index. We present further the relations between general Sombor index and other generalised indices: general Randić index and general sum-connectivity index.
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The Sombor and reduced Sombor indices of $G$ are defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{deg_G(u)^2+deg_G(v)^2}$ and $SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(deg_G(u)-1)^2+(deg_G(v)-1)^2}$, respectively. We denote by $H_{n,ν}$ the graph constructed from the star $S_n$ by adding $ν$ edge(s) $(0\leq ν\leq n-2)$, 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, $\textit{Contrib. Math. }$ $\textbf{3}$ (2021) 11-18] proposed a conjecture that the graph…
The Sombor index (SO) is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices of $\sqrt{d_i^2+d_j^2}$, where $d_i$ is the degree of the $i$-th vertex. It has been conceived using geometric considerations. Recently, a series of new SO-like degree-based graph invariants (denoted by $SO_1, SO_2,..., SO_6$) is taken into consideration, when the geometric background of several classical topological indices (Zagreb, Albertson) has considered. In this paper, we compute and study these new indices for some graphs, cactus chains and polymers.
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 pairs of non-adjacent vertices, capturing, thus, and quantifying a possible influence of remote pairs of vertices. We give several properties of the Somber coindex and its relations to the Sombor index, the Zagreb (co)indices, forgotten coindex and other important graph parameters. We also compute the bounds of the Somber coindex of some graph operations and …
Let $d_v$ be the degree of the vertex $v$ in a connected graph $G$. The first Banhatti-Sombor index of $G$ is defined as $BSO(G) =\sum_{uv\in E(G)}\sqrt{\frac{1}{d^2_u}+\frac{1}{d^2_v}}$, which is a new vertex-degree-based topological index introduced by Kulli. In this paper, the mathematical relations between the first Banhatti-Sombor index and some other well-known vertex-degree-based topological indices are established. In addition, the trees extremal with respect to the first Banhatti-Sombor index on trees and chemical trees are characterized, respectively.
A new geometric background of graph invariants was introduced by Gutman, of which the simplest is the second Sombor index $SO_2$, defined as $SO_2=SO_2(G)=\sum_{uv\in E}\frac{|d^2_G(u)-d^2_G(v)|}{d^2_G(u)+d^2_G(v)}$, where $G = (V, E)$ is a simple graph and $d_G(v)$ denotes the degree of $v$ in $G$. In this paper, the chemical applicability of the second Sombor index is investigated and it is shown that the the second Sombor index is useful in predicting physicochemical properties with high accuracy compared to some well-established and often used indices. Also, we obtain a bound for the secon…
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 graphs with k cut-vertices. In this paper, we present minimum Sombor indices of graphs in G^k_n. The corresponding extremal graphs have been characterized as well.
Vertex-degree-based topological indices have recently gained a lot of attention from mathematical chemists. One such index that we focus on in this paper is called Sombor index. After its definition in late 2020, the Sombor index was quickly recognized as a valuable research topic. In this paper we partially answer the open question of finding the extremal trees with respect to this index for a fixed degree sequence. Particularly we focus on the lower bound and proceed to show that greedy tree minimizes the Sombor index for a given degree sequence.
We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index: $SP_α(G) = \sum_{uv \in E(G)} d_u$, if $d_u=d_v$, and $SP_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α-d_v^α\right)/\left( α(d_u-d_v\right)\right]^{1/(α-1)}$, otherwise. Here, $uv$ denotes the edge of the network $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0,1\}$. Indeed, for given values of $α$, the Stolarsky-Puebla index reproduces…
The signless Laplacian matrix in graph spectra theory is a remarkable matrix of graphs, and it is extensively studied by researchers. In 1981, Cvetković pointed $12$ directions in further investigations of graph spectra, one of which is "classifying and ordering graphs". Along with this classic direction, we pay our attention on the order of the largest eigenvalue of the signless Laplacian matrix of graphs, which is usually called the $Q$-index of a graph. Let $\mathbb{G}(m, g)$ (resp. $\mathbb{G}(m, \geq g)$) be the family of connected graphs on $m$ edges with girth $g$ (resp. no less than $g…
The largest $n-1$ Hosoya indices of unicyclic graphs
图$G$的Hosoya指标$Z (G)$定义为图$G$的边独立集的总数.本文推广了[J. Ou，On extremal unicyclic molecular graphs with maximum Hosoya index，\textit{Discrete Appl. Math.} 157（2009）391--397.]和[Y。是的，X。潘，H. Liu，Ordering unicyclic graphs with respect to Hosoya indices and Merrifield-Simmons indices，\textit{MATCH Commun. Math. Comput. Chem.} 59（2008）191--202.]并根据Hosoya指标对最大的$n-1$单圈图进行排序。
For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{i}$ the degree of vertex $v_{i}$. Let $f(x, y)>0$ be a real symmetric function in $x$ and $y$. The weighted adjacency matrix $A_{f}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle f(d_{i}, d_{j})$ if the vertices $v_{i}$ and $v_{j}$ are adjacent and 0 otherwise. Li and Wang \cite{U9} tried to unify methods to study spectral radius of weighted adjacency matrices of graphs weighted by various topological indices. If $\displaystyle f'_{x}(x, y)\geq0$ and $\displaystyle f''_{x}(x, y)\geq0$, then $\displ…
