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Line chart showing the population of the town of Pushkin, Saint Petersburg from 1800 to 2010, measured at various intervals. A line chart or line graph, also known as curve chart, [1] is a type of chart that displays information as a series of data points called 'markers' connected by straight line segments. [2]
In a line graph L(G), each vertex of degree k in the original graph G creates k(k − 1)/2 edges in the line graph. For many types of analysis this means high-degree nodes in G are over-represented in the line graph L(G). For instance, consider a random walk on the vertices of the original graph G.
Graph.Edges(u, v) returns the length of the edge joining (i.e. the distance between) the two neighbor-nodes u and v. The variable alt on line 14 is the length of the path from the source node to the neighbor node v if it were to go through u. If this path is shorter than the current shortest path recorded for v, then the distance of v is ...
(A 2-uniform hypergraph is a graph). In hypergraph theory, it is often natural to require that hypergraphs be k-uniform. Every graph is the line graph of some hypergraph, but, given a fixed edge size k, not every graph is a line graph of some k-uniform hypergraph. A main problem is to characterize those that are, for each k ≥ 3.
A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest . Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts.
The [5 dl 2 dl 1 dl 2 dl] represents a dash-dot line. There are 5 units of line (the dash) followed by 2 units of empty space, 1 unit of line (the dot), 2 more units of empty space, and then it starts over again. 0.5 0.5 0.5 represents the color gray. /LTb is the graph's border, and /LTa is for the zero axes. [9]
In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple cycle is a triangle. [1] A graph is line perfect if and only if each of its biconnected components is a bipartite graph, the complete graph K 4, or a triangular book K 1,1,n. [2]
Examples include the rook's graphs, the line graphs of complete bipartite graphs. Every line graph of a bipartite graph is an induced subgraph of a rook's graph. [20] Because line graphs of bipartite graphs are perfect, their clique number equals their chromatic number. The clique number of the line graph of a bipartite graph is the maximum ...