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The intersection number arises in the study of fixed points, which can be cleverly defined as intersections of function graphs with a diagonals. Calculating the intersection numbers at the fixed points counts the fixed points with multiplicity , and leads to the Lefschetz fixed-point theorem in quantitative form.
A graph with intersection number four. The four shaded regions indicate cliques that cover all the edges of the graph. An alternative definition of the intersection number of a graph is that it is the smallest number of cliques in (complete subgraphs of ) that together cover all of the edges of .
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and the distance between v and w. Some authors exclude the complete graphs and disconnected graphs from this definition.
The line graph of a graph G is defined as the intersection graph of the edges of G, where we represent each edge as the set of its two endpoints. A string graph is the intersection graph of curves on a plane. A graph has boxicity k if it is the intersection graph of multidimensional boxes of dimension k, but not of any smaller dimension.
Klein & Margraf (2003) define the linear intersection number of a graph, similarly, to be the minimum number of vertices in a linear hypergraph whose line graph is G. As they observe, the Erdős–Faber–Lovász conjecture is equivalent to the statement that the chromatic number of any graph is at most equal to its linear intersection number.
Graph = with the -axis as the horizontal axis and the -axis as the vertical axis.The -intercept of () is indicated by the red dot at (=, =).. In analytic geometry, using the common convention that the horizontal axis represents a variable and the vertical axis represents a variable , a -intercept or vertical intercept is a point where the graph of a function or relation intersects the -axis of ...
Average mortgage rates for popular 30-year fixed terms are relatively flat while shorter 15-year fixed terms tick up as of Wednesday, January 8, 2025, elevated to their highest levels in six months.
Intersection graphs An interval graph is the intersection graph of a set of line segments in the real line. It may be given an adjacency labeling scheme in which the points that are endpoints of line segments are numbered from 1 to 2n and each vertex of the graph is represented by the numbers of the two endpoints of its corresponding interval ...
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