Search Results (4)

In this paper, motivated by the recently introduced topological indices—the Sombor index, Sombor coindex, and Lanzhou index, we define a new index—the Lanzhou coindex of a graph. Furthermore, we investigate the Sombor index (coindex) and the Lanzhou index (coindex) of tadpole graphs, wheel graphs, and two-dimensional grid graphs, as well as their paraline graphs. Full article

This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set $\{x_1,\dots ,x_n\}$, then the ad-hoc Lanzhou index of G is defined by $\widetilde{Lz}\left(G\right)={\sum }_{i=1}^n{d}_i{(n-1-d_i)}^2$, where $d_i$ represents the degree of the vertex $x_i$. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected $\xi$-cyclic graphs of a fixed order, (ii) the set of all connected molecular $\xi$-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order. Full article

A topological index being a graph theoretic parameter plays a role of function for the assignment of a numerical value to a molecular graph which predicts the several physical and chemical properties of the underlying molecular graph such as heat of evaporation, critical temperature, surface tension, boiling point, octanol-water partition coefficient, density and flash points. For a (molecular) graph $\mathsf{\Gamma }$, the Lanzhou index (Lz index) is obtained by the sum of $deg{\left(v\right)}^2$$d\overline{e}g\left(v\right)$ over all the vertices, where $deg\left(v\right)$ and $d\overline{e}g\left(v\right)$ are degrees of the vertex v in $\mathsf{\Gamma }$ and its complement $\overline{\mathsf{\Gamma }}$ respectively. Let ${\mathcal{V}}_{\alpha }^{\beta }$ be a class of unicyclic graphs (same order and size) such that each graph of this class has order $\alpha$ and $\beta$ leaves (vertices of degree one). In this note, we compute the lower and upper bounds of Lz index for each unicyclic graph in the class of graphs ${\mathcal{V}}_{\alpha }^{\beta }.$ Moreover, we characterize the extremal graphs with respect to Lz index in the same class of graphs. Full article

As the foundation and skeleton of urban space, the street network is significant to the urban travel environment and socio-economic activities. To reveal the structural characteristics of the street network, this paper proposes a measurement index system to study the street network structure and urban travel characteristics. To illustrate the relationship between spatial accessibility of streets in strip cities and residents’ travel and service demands, we take Lanzhou, a typical strip city, as an example for network analysis and study the hierarchical structure of physical, functional, and environmental characteristics of the street topological network. The results show that Lanzhou City has formed a radial network structure with traffic-oriented streets as the backbone and interconnected living streets. However, the development of old and new urban areas is still uneven. In terms of street function distribution, streets with a high degree of diversity are more attractive to population clustering and show a polycentric clustering feature in space related to the regional functional orientation and travel characteristics. Much of the structural difference in the centrality core-periphery of the street network under pedestrian and vehicular travel patterns are influenced by the street’s type and function. In addition, as part of the contribution, we provide an evaluation methodology that enables the analysis of street network centrality. These findings advance our understanding of strip city development. Full article