Category: Algebraic graph theory

Alon–Boppana bound
In spectral graph theory, the Alon–Boppana bound provides a lower bound on the second-largest eigenvalue of the adjacency matrix of a -regular graph, meaning a graph in which every vertex has degree .
Alpha centrality
In graph theory and social network analysis, alpha centrality is an alternative name for Katz centrality. It is a measure of centrality of nodes within a graph. It is an adaptation of eigenvector cent
Dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separ
Parry–Sullivan invariant
In mathematics, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices in graph theory, and of certain one-dimensional dynamical
Lovász conjecture
In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally Lás
Conductance (graph)
In graph theory the conductance of a graph G = (V, E) measures how "well-knit" the graph is: it controls how fast a random walk on G converges to its stationary distribution. The conductance of a grap
Hafnian
In mathematics, the hafnian of an adjacency matrix of a graph is the number of perfect matchings in the graph. It was so named by Eduardo R. Caianiello "to mark the fruitful period of stay in Copenhag
Semi-symmetric graph
In the mathematical field of graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. In other words, a graph is semi-symmetric if ea
Spectral clustering
In multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer di
Integral graph
In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the root
Mac Lane's planarity criterion
In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane, who published it in 1937. It states that a finite
Complex network zeta function
Different definitions have been given for the dimension of a complex network or graph. For example, metric dimension is defined in terms of the resolving set for a graph. Dimension has also been defin
Kirchhoff's theorem
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
Ramanujan graph
In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectr
Tutte matrix
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If
Centrality
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most inf
Edmonds matrix
In graph theory, the Edmonds matrix of a balanced bipartite graph with sets of vertices and is defined by where the xij are indeterminates. One application of the Edmonds matrix of a bipartite graph i
Vertex-transitive graph
In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism such that In other words, a graph is verte
Distance-regular graph
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 depe
Matching polynomial
In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a
Seidel adjacency matrix
In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whos
Minimum rank of a graph
In mathematics, the minimum rank is a graph parameter for a graph G. It was motivated by the Colin de Verdière graph invariant.
Laplacian matrix
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Name
Cycle space
In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs. This set of subgraphs can be described algebraically as a vector spac
Symmetric graph
In the mathematical field of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphism such that and In other
Walk-regular graph
In discrete mathematics, a walk-regular graph is a simple graph where the number of closed walks of any length from a vertex to itself does not depend on the choice of vertex.
Conference graph
In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, k = (v − 1)/2, λ = (v − 5)/4, and μ = (v − 1)/4. It is the graph associated with a symmetric
Modularity (networks)
Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modul
Algebraic graph theory
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are
Graham–Pollak theorem
In graph theory, the Graham–Pollak theorem states that the edges of an -vertex complete graph cannot be partitioned into fewer than complete bipartite graphs. It was first published by Ronald Graham a
Incidence matrix
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is X and the second is Y, t
Conference matrix
In mathematics, a conference matrix (also called a C-matrix) is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I. Thus, if
Elementary Number Theory, Group Theory and Ramanujan Graphs
Elementary Number Theory, Group Theory and Ramanujan Graphs is a book in mathematics whose goal is to make the construction of Ramanujan graphs accessible to undergraduate-level mathematics students.
Clustering coefficient
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 ne
Graph energy
In mathematics, the energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. This quantity is studied in the context of spectral graph theory. More
Edge and vertex spaces
In the mathematical discipline of graph theory, the edge space and vertex space of an undirected graph are vector spaces defined in terms of the edge and vertex sets, respectively. These vector spaces
Adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the
Adjacency algebra
In algebraic graph theory, the adjacency algebra of a graph G is the algebra of polynomials in the adjacency matrix A(G) of the graph. It is an example of a matrix algebra and is the set of the linear
Two-graph
In mathematics, a two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple from X contains an even number of triples of the two-graph. A regul
Graph automorphism
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automor
Degree matrix
In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edg
Rank (graph theory)
In graph theory, a branch of mathematics, the rank of an undirected graph has two unrelated definitions. Let n equal the number of vertices of the graph. * In the matrix theory of graphs the rank r o
Hierarchical closeness
Hierarchical closeness (HC) is a structural centrality measure used in network theory or graph theory. It is extended from closeness centrality to rank how centrally located a node is in a directed ne
Cycle basis
In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that
Algebraic connectivity
The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the L
Frucht's theorem
Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite
Half-transitive graph
In the mathematical field of graph theory, a half-transitive graph is a graph that is both vertex-transitive and edge-transitive, but not symmetric. In other words, a graph is half-transitive if its a
Spectral graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, s
Jordan–Pólya number
In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, is a Jordan–Pól
Babai's problem
In algebraic graph theory, Babai's problem was proposed in 1979 by László Babai.
Zero-symmetric graph
In the mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique symmetry taki
Strongly regular graph
In graph theory, a strongly regular graph (SRG) is defined as follows. Let G = (V, E) be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and
Brouwer's conjecture
In the mathematical field of spectral graph theory, Brouwer's conjecture is a conjecture by Andries Brouwer on upper bounds for the intermediate sums of the eigenvalues of the Laplacian of a graph in
Distance-transitive graph
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance,
Sims conjecture
In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims. He conjectured that if is a primitive permutation group on a finite set and denotes the stabilizer
Ihara zeta function
In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adj
Edge-transitive graph
In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. In other words, a grap