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Register a new userMaximum Sombor index among cacti
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Hechao Liu
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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.
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The edge Szeged index and edge-vertex Szeged index of a graph are defined as $Sz_{e}(G)=sumlimits_{uvin E(G)}m_{u}(uv|G)m_{v}(uv|G)$ and $Sz_{ev}(G)=frac{1}{2} sumlimits_{uv in E(G)}[n_{u}(uv|G)m_{v}(uv|G)+n_{v}(uv|G)m_{u}(uv|G)],$ respectively, where $m_{u}(uv|G)$ (resp., $m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $n_{u}(uv|G)$ (resp., $n_{v}(uv|G)$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), respectively. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, the lower bounds of edge Szeged index and edge-vertex Szeged index for cacti with order $n$ and $k$ cycles are determined, and all the graphs that achieve the lower bounds are identified.
Let $G$ be a connected graph. The edge revised Szeged index of $G$ is defined as $Sz^{ast}_{e}(G)=sumlimits_{e=uvin E(G)}(m_{u}(e|G)+frac{m_{0}(e|G)}{2})(m_{v}(e|G)+frac{m_{0}(e|G)}{2})$, where $m_{u}(e|G)$ (resp., $m_{v}(e|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $m_{0}(e|G)$ is the number of edges equidistant from both ends of $e$. In this paper, we give the minimal and the second minimal edge revised Szeged index of cacti with order $n$ and $k$ cycles, and all the graphs that achieve the minimal and second minimal edge revised Szeged index are identified.
The edge-Wiener index $W_e(G)$ of a connected graph $G$ is the sum of distances between all pairs of edges of $G$. A connected graph $G$ is said to be a cactus if each of its blocks is either a cycle or an edge. Let $mathcal{G}_{n,t}$ denote the class of all cacti with $n$ vertices and $t$ cycles. In this paper, the upper bound and lower bound on the edge-Wiener index of graphs in $mathcal{G}_{n,t}$ are identified and the corresponding extremal graphs are characterized.
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.
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.
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