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In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points ) which are connected by edges (also called arcs , links or lines ).
Spectral graph theory relates properties of a graph to a spectrum, i.e., eigenvalues and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. Imbalanced weights may undesirably affect the matrix spectrum, leading to the need of normalization — a column/row scaling of the matrix entries ...
In graph theory, a deletion-contraction formula / recursion is any formula of the following recursive form: = + (/). Here G is a graph, f is a function on graphs, e is any edge of G, G \ e denotes edge deletion, and G / e denotes contraction. Tutte refers to such a function as a W-function. [1]
An undirected graph. In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented.. The unoriented incidence matrix (or simply incidence matrix) of an undirected graph is a matrix B, where n and m are the numbers of vertices and edges respectively, such that
One can define the adjacency matrix of a directed graph either such that a non-zero element A ij indicates an edge from i to j or; it indicates an edge from j to i. The former definition is commonly used in graph theory and social network analysis (e.g., sociology, political science, economics, psychology). [5]
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established ...
In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix.
PARI/GP is a computer algebra system that facilitates number-theory computation. Besides support of factoring, algebraic number theory, and analysis of elliptic curves, it works with mathematical objects like matrices, polynomials, power series, algebraic numbers, and transcendental functions. [3]