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The windows are therefore stacked along the Z-axis, and the Z-order information thus specifies the front-to-back ordering of the windows on the screen. An analogy would be some sheets of paper scattered on top of a table, each sheet being a window, the table your computer screen, and the top sheet having the highest Z value.
The complete graph K 4 has the ten matchings shown, so its Hosoya index is ten, the maximum for any four-vertex graph. The Hosoya index, also known as the Z index, of a graph is the total number of matchings in it. The Hosoya index is always at least one, because the empty set of edges is counted as a matching for this purpose. Equivalently ...
Figure 2. Box-plot with whiskers from minimum to maximum Figure 3. Same box-plot with whiskers drawn within the 1.5 IQR value. A boxplot is a standardized way of displaying the dataset based on the five-number summary: the minimum, the maximum, the sample median, and the first and third quartiles.
The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. [5] [6] The basic idea is to sort the input set according to Z-order.Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, [7] or they can be used to build a pointer based quadtree.
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For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G.The list coloring number ch(G) satisfies the following properties.. ch(G) ≥ χ(G).A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring.
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Every graph has a cycle basis in which every cycle is an induced cycle. In a 3-vertex-connected graph, there always exists a basis consisting of peripheral cycles, cycles whose removal does not separate the remaining graph. [4] [5] In any graph other than one formed by adding one edge to a cycle, a peripheral cycle must be an induced cycle.