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where the degree of a vertex counts the number of times an edge terminates at that vertex. In an undirected graph , this means that each loop increases the degree of a vertex by two. In a directed graph , the term degree may refer either to indegree (the number of incoming edges at each vertex) or outdegree (the number of outgoing edges at ...
The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; [5] for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant, so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non ...
In set theory and graph theory, denotes the set of n-tuples of elements of , that is, ordered sequences of elements that are not necessarily distinct. In the edge ( x , y ) {\displaystyle (x,y)} directed from x {\displaystyle x} to y {\displaystyle y} , the vertices x {\displaystyle x} and y {\displaystyle y} are called the endpoints of the ...
The Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in many areas: for example, constructing well-connected networks of computers , card shuffling , and low-dimensional ...
Its authors have divided Elementary Number Theory, Group Theory and Ramanujan Graphs into four chapters. The first of these provides background in graph theory, including material on the girth of graphs (the length of the shortest cycle), on graph coloring, and on the use of the probabilistic method to prove the existence of graphs for which both the girth and the number of colors needed are ...
In a weighted graph, a vertex may have a large degree because of a small number of connected edges but with large weights just as well as due to a large number of connected edges with unit weights. Graph self-loops, i.e., non-zero entries on the main diagonal of the adjacency matrix, do not affect the graph Laplacian values, but may need to be ...
In graph theory, a bipolar orientation or st-orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that causes the graph to become a directed acyclic graph with a single source s and a single sink t, and an st-numbering of the graph is a topological ordering of the resulting directed acyclic graph.
The degree diameter problem seeks tight relations between the diameter, number of vertices, and degree of a graph. One way of formulating it is to ask for the largest graph with given bounds on its degree and diameter. For any fixed degree, this maximum size is exponential in the diameter, with the base of the exponent depending on the degree. [1]