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The number of these irreducibles is equal to the number of conjugacy classes of G. The above fact can be explained by character theory. Recall that the character of the regular representation χ(g) is the number of fixed points of g acting on the regular representation V. It means the number of fixed points χ(g) is zero when g is not id and |G ...
The condensation of a directed graph G is a directed acyclic graph with one vertex for each strongly connected component of G, and an edge connecting pairs of components that contain the two endpoints of at least one edge in G. cone A graph that contains a universal vertex. connect Cause to be connected. connected
Following the terminology in much of the strongly regular graph literature, the larger eigenvalue is called r with multiplicity f and the smaller one is called s with multiplicity g. Since the sum of all the eigenvalues is the trace of the adjacency matrix, which is zero in this case, the respective multiplicities f and g can be calculated:
As a diagram this is a singleheaded arrow. Symmetrically, the corresponding bra is j, m|. In diagram form this is a doubleheaded arrow, pointing in the opposite direction to the ket. In each case; the quantum numbers j, m are often labelled next to the arrows to refer to a specific angular momentum state,
Many problems and theorems in graph theory have to do with various ways of coloring graphs. Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. One may also consider coloring edges (possibly so that no two coincident edges are the same color), or other ...
One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"—if there is a bijection between them.
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows. The objects are the elements of G , and for any two elements x and y of G , there exists a unique morphism from x to y if and only if x ∼ y . {\displaystyle x\sim y.}
where | g | is the absolute value of the determinant of the matrix of scalar coefficients of the metric tensor . These are useful when dealing with divergences and Laplacians (see below). The covariant derivative of a vector field with components is given by: