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In the special case when k is the function field of an algebraic curve over a finite field and f is any character that is trivial on k, this recovers the geometric Riemann–Roch theorem. [12] Other versions of the arithmetic Riemann–Roch theorem make use of Arakelov theory to resemble the traditional Riemann–Roch theorem more exactly.
The Rankine scale is used in engineering systems where heat computations are done using degrees Fahrenheit. [3] The symbol for degrees Rankine is °R [2] (or °Ra if necessary to distinguish it from the Rømer and Réaumur scales). By analogy with the SI unit kelvin, some authors term the unit Rankine, omitting the degree symbol. [4] [5]
(For a graph with n vertices and r terminals, they use t = n − r − 1 added vertices per tree.) Then, they ask for the k -minimum spanning tree in this augmented graph with k = rt . The only way to include this many vertices in a k -spanning tree is to use at least one vertex from each added tree, for there are not enough vertices remaining ...
From the handshaking lemma, a k-regular graph with odd k has an even number of vertices. A theorem by Nash-Williams says that every k ‑regular graph on 2k + 1 vertices has a Hamiltonian cycle. Let A be the adjacency matrix of a graph. Then the graph is regular if and only if = (, …,) is an eigenvector of A. [2]
A complete graph (denoted , where is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, . In a signed graph , the number of positive edges connected to the vertex v {\displaystyle v} is called positive deg ( v ) {\displaystyle (v)} and the number of connected negative ...
The size of G is bounded above by the Moore bound; for 1 < k and 2 < d, only the Petersen graph, the Hoffman-Singleton graph, and possibly graphs (not yet proven to exist) of diameter k = 2 and degree d = 57 attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound.
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Some examples are the even cycles C 2n, the complete bipartite graphs K n,n with girth four, the Heawood graph with degree 3 and girth 6, and the Tutte–Coxeter graph with degree 3 and girth 8. More generally it is known that, other than the graphs listed above, all Moore graphs must have girth 5, 6, 8, or 12. [6]