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Register a new userThe expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs
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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.
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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 $$SO(G)=sum_{uvin E(G)}sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)},$$ $$SO_{red}(G)=sum_{uvin E(G)}sqrt{(d_{G}(u)-1)^{2}+(d_{G}(v)-1)^{2}},$$ 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.
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: Erdos-Renyi 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 are highly correlated with the Shannon entropy of the eigenvectors of the graph adjacency matrix.
Sombor index is a novel topological index introduced by Gutman, defined as $SO(G)=sumlimits_{uvin E(G)}sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}$ denotes the degree of vertex $u$. Recently, Chen et al. [H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022), in press] considered the Sombor indices of trees with given diameter. For the continue, we determine the maximum Sombor indices for unicyclic graphs with given diameter.
Recently, a novel topological index, Sombor index, was introduced by Gutman, defined as $SO(G)=sumlimits_{uvin E(G)}sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}$ denotes the degree of vertex $u$. In this paper, we first determine the maximum Sombor index among cacti with $n$ vertices and $t$ cycles, then determine the maximum Sombor index among cacti with perfect matchings. We also characterize corresponding maximum cacti.
Let $G=(V(G),E(G))$ be a simple graph with vertex set $V(G)={v_{1},v_{2},cdots, v_{n}}$ and edge set $E(G)$. The $p$-Sombor matrix $mathcal{S}_{p}(G)$ of $G$ is defined as the $(i,j)$ entry is $((d_{i})^{p}+(d_{j})^{p})^{frac{1}{p}}$ if $v_{i}sim v_{j}$, and 0 otherwise, where $d_{i}$ denotes the degree of vertex $v_{i}$ in $G$. In this paper, we study the relationship between $p$-Sombor index $SO_{p}(G)$ and $p$-Sombor matrix $S_{p}(G)$ by the $k$-th spectral moment $N_{k}$ and the spectral radius of $S_{p}(G)$. Then we obtain some bounds of $p$-Sombor Laplacian spectrum, $p$-Sombor spectral radius, $p$-Sombor spectral spread, $p$-Sombor energy and $p$-Sombor Estrada index. We also investigate the Nordhaus-Gaddum-type results for $p$-Sombor spectral radius and energy. At last, we give the regression model for boiling point and some other invariants.
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