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Graph Polynomial
In mathematics, a graph polynomial is a graph invariant whose value is a polynomial. Invariants of this type are studied in algebraic graph theory. Important graph polynomials include: *The characteristic polynomial, based on the graph's adjacency matrix. *The chromatic polynomial, a polynomial whose values at integer arguments give the number of colorings of the graph with that many colors. *The dichromatic polynomial, a 2-variable generalization of the chromatic polynomial *The flow polynomial, a polynomial whose values at integer arguments give the number of nowhere-zero flows with integer flow amounts modulo the argument. *The (inverse of the) Ihara zeta function, defined as a product of binomial terms corresponding to certain closed walks in a graph. *The Martin polynomial, used by Pierre Martin to study Euler tours *The matching polynomials, several different polynomials defined as the generating function of the matchings of a graph. *The reliability polynomial, a polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Property
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph.. Definitions While graph drawing and graph representation are valid topics in graph theory, in order to focus only on the abstract structure of graphs, a graph property is defined to be a property preserved under all possible isomorphisms of a graph. In other words, it is a property of the graph itself, not of a specific drawing or representation of the graph. Informally, the term "graph invariant" is used for properties expressed quantitatively, while "property" usually refers to descriptive characterizations of graphs. For example, the statement "graph does not have vertices of degree 1" is a "property" while "the number of vertices of degree 1 in a graph" is an "invariant". More formally, a graph property is a class of graphs with the property that any two isomorphic gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Tour
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this: :Given the graph in the image, is it possible to construct a path (or a cycle; i.e., a path starting and ending on the same vertex) that visits each edge exactly once? Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: :A con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Polynomial
In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. History The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later. In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial. The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial. Soon after Jones' discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a partition function (state-sum model), which involved the bracket polynomial, an invariant of framed knots. This opened up avenues of research linki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges, from the original graph, connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subseteq V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. * Induced paths are induced subgraphs that are paths. The shortest path between any two vertices in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte Polynomial
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contains information about how the graph is connected. It is denoted by T_G. The importance of this polynomial stems from the information it contains about G. Though originally studied in algebraic graph theory as a generalization of counting problems related to graph coloring and nowhere-zero flow, it contains several famous other specializations from other sciences such as the Jones polynomial from knot theory and the partition functions of the Potts model from statistical physics. It is also the source of several central computational problems in theoretical computer science. The Tutte polynomial has several equivalent definitions. It is essentially equivalent to Whitney’s rank polynomial, Tutte’s own dichromatic polynomial and Fortuin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reliability Polynomial
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contains information about how the graph is connected. It is denoted by T_G. The importance of this polynomial stems from the information it contains about G. Though originally studied in algebraic graph theory as a generalization of counting problems related to graph coloring and nowhere-zero flow, it contains several famous other specializations from other sciences such as the Jones polynomial from knot theory and the partition functions of the Potts model from statistical physics. It is also the source of several central computational problems in theoretical computer science. The Tutte polynomial has several equivalent definitions. It is essentially equivalent to Whitney’s rank polynomial, Tutte’s own dichromatic polynomial and Fortuin†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching (graph Theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected Graph (discrete mathematics), graph is a set of Edge (graph theory), edges without common vertex (graph theory), vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a Flow network, network flow problem. Definitions Given a Graph (discrete mathematics), graph a matching ''M'' in ''G'' is a set of pairwise non-adjacent edges, none of which are loop (graph theory), loops; that is, no two edges share common vertices. A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated). A maximal matching is a matching ''M'' of a graph ''G'' that is not a subset of any other matching. A matching ''M'' of a graph ''G'' is maximal if every edge in ''G'' has a non-empty intersectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 graph. It is one of several graph polynomials studied in algebraic graph theory. Definition Several different types of matching polynomials have been defined. Let ''G'' be a graph with ''n'' vertices and let ''mk'' be the number of ''k''-edge matchings. One matching polynomial of ''G'' is :m_G(x) := \sum_ m_k x^k. Another definition gives the matching polynomial as :M_G(x) := \sum_ (-1)^k m_k x^. A third definition is the polynomial :\mu_G(x,y) := \sum_ m_k x^k y^. Each type has its uses, and all are equivalent by simple transformations. For instance, :M_G(x) = x^n m_G(-x^) and :\mu_G(x,y) = y^n m_G(x/y^2). Connections to other polynomials The first type of matching polynomial is a direct generalization of the rook polynomial. The second type of matching polynomial has remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Polynomial
Martin may refer to: Places Antarctica * Martin Peninsula, Marie Byrd Land * Port Martin, Adelie Land * Point Martin, South Orkney Islands Europe * Martin, Croatia, a village * Martin, Slovakia, a city * MartÃn del RÃo, Aragón, Spain * MartÃn River, a tributary of the Ebro river in Spain * Martin (Val Poschiavo), Switzerland England * Martin, Hampshire * Martin, Kent * Martin, East Lindsey, Lincolnshire, a hamlet and former parish * Martin, North Kesteven, Lincolnshire, a village and parish * Martin Hussingtree, Worcestershire * Martin Mere, a lake in Lancashire ** WWT Martin Mere, a wetland nature reserve that includes the lake and surrounding areas North America Canada * Rural Municipality of Martin No. 122, Saskatchewan, Canada * Martin Islands, Nunavut, Canada United States * Martin, Florida * Martin, Georgia * Martin, Indiana * Martin, Kentucky * Martin, Louisiana * Martin, Michigan * Martin, Nebraska * Martin, North Dakota * Martin, Ohio * Martin, South Carolina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 adjacency matrix. The Ihara zeta function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear group. Jean-Pierre Serre suggested in his book ''Trees'' that Ihara's original definition can be reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice in 1985. As observed by Sunada, a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis. Definition The Ihara zeta function is defined as the analytic continuation of the infinite product :\zeta_(u)=\prod_\frac, where ''L''(''p'') is the ''length'' L(p) of p. The product in the definition is taken over all prime closed geodesics p of the graph G = (V, E), whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
