Search Results (6)

Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems. Full article

A connected graph, G, is Crossing Free-connected ($\mathcal{CF}$-connected) if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G. We conjecture that a complete tripartite graph, $K_{l,m,n}$, is $\mathcal{CF}$-connected if and only if it does not contain any of the following as a subgraph: $K_{1,2,7}$, $K_{1,3,5}$, $K_{1,4,4}$, $K_{2,2,5}$, $K_{3,3,3}$. We examine the idea that $K_{1,2,7}$, $K_{1,3,5}$, $K_{1,4,4}$, and $K_{2,2,5}$ are the first non-$\mathcal{CF}$-connected complete tripartite graphs. The $\mathcal{CF}$-connectedness of $K_{l,m,n}$ with $l,m,n\ge 3$ is dependent on the knowledge of crossing numbers of $K_{3,3,n}$. In this paper, we prove various results that support this conjecture. Full article

Reducing the number of crossings on graph edges can be useful in various applications, including network visualization, circuit design, graph theory, cartography or social choice theory. This paper aims to determine the crossing number of the join product $G^{*}+D_n$, where $G^{*}$ is a connected graph isomorphic to $K_{2,2,2}\setminus \{e_1,e_2\}$ obtained by removing two edges $e_1,e_2$ with a common vertex and a second vertex from the different partitions of the complete tripartite graph $K_{2,2,2}$, and $D_n$ is a discrete graph composed of n isolated vertices. The proofs utilize known exact crossing number values for join products of specific subgraphs $H_k$ of $G^{*}$ with discrete graphs in combination with the separating cycles. Similar approaches can potentially estimate unknown crossing numbers of other six-vertex graphs with a larger number of edges in join products with discrete graphs, paths or cycles. Full article

For a graph $G=(V,E)$ and a set $S\subseteq V\left(G\right)$ of a size at least 2, a path in G is said to be an S-path if it connects all vertices of S. Two S-paths $P_1$ and $P_2$ are said to be internally disjoint if $E\left(P_1\right)\cap E\left(P_2\right)=\varnothing$ and $V\left(P_1\right)\cap V\left(P_2\right)=S$; that is, they share no vertices and edges apart from S. Let ${\pi }_G\left(S\right)$ denote the maximum number of internally disjoint S-paths in G. The k-path-connectivity ${\pi }_k\left(G\right)$ of G is then defined as the minimum ${\pi }_G\left(S\right)$, where S ranges over all k-subsets of $V\left(G\right)$. In this paper, we study the k-path-connectivity of the complete balanced tripartite graph $K_{n,n,n}$ and obtain ${\pi }_k\left(K_{n,n,n}\right)=⌊\frac{2n}{k-1}⌋$ for $3\le k\le n.$ Full article

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph $K_{1,1,3}\backslash e$ obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph $K_{1,1,3}$. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices. Full article

Points of interest (POIs) such as stores, gas stations, and parking lots are particularly important for maps. Using gas station as a case study, this paper proposed a novel approach to enhance POI information using low-frequency vehicle trajectory data and social media data. First, the proposed method extracted spatial information of the gas station from sparse vehicle trace data in two steps. The first step proposed the velocity sequence linear clustering algorithm to extract refueling stop tracks from the individual trace line after modeling the vehicle refueling stop behavior using movement features. The second step used the Delaunay triangulation to extract the spatial information of gas stations from the collective refueling stop tracks. Second, attribute information and dimension sentiment semantic information of the gas station were extracted from social media data using the text mining method and tripartite graph model. Third, the gas station information was enhanced by fusing the extracted spatial data and semantic data using a matching method. Experiments were conducted using the 15-day vehicle trajectories of 12,000 taxis and social media data from the Dazhongdianping in Beijing, China, and the results showed that the proposed method could extract the spatial information, attribute information, and review information of gas stations simultaneously. Compared with ground truth data, the automatically enhanced gas station was proved to be of higher quality in terms of the correctness, completeness, and real-time. Full article