Signed graph

In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.

Formally, a signed graph Σ is a pair ("G", σ) that consists of a graph "G" = ("V", "E") and a sign mapping or signature σ from "E" to the sign group {+,−}. The graph may have loops and multiple edges as well as half-edges (with only one endpoint) and loose edges (with no endpoints). Half and loose edges do not receive signs. (In the terminology of the article on graphs, it is a "multigraph", but we say "graph" because in signed graph theory it is usually unnatural to restrict to simple graphs.)The sign of a circle (this is the edge set of a simple cycle) is defined to be the product of the signs of its edges; in other words, a circle is positive if it contains an even number of negative edges and negative if it contains an odd number of negative edges. The fundamental fact about a signed graph is the list of positive circles, which we write "B"(Σ). A signed graph, or a subgraph or edge set, is called balanced if every circle in it is positive (and it contains no half-edges). Two fundamental questions about a signed graph are: Is it balanced? What is the largest size of a balanced edge set in it? The first question is not difficult; the second is computationally intractable (technically, it is NP-hard).

Signed graphs were first introduced by Harary to handle a problem in social psychology (Cartwright and Harary, 1956). They have been rediscovered many times because they come up naturally in many unrelated areas. For instance, they enable one to describe and analyze the geometry of subsets of the classical root systems. They appear in topological graph theory and group theory. They are a natural context for questions about odd and even cycles in graphs. They appear in computing the ground state energy in the non-ferromagnetic Ising model; for this one needs to find a largest balanced edge set in Σ.

Examples

* The complete signed graph on "n" vertices with loops, denoted by ±"K"_"n"^o, has every possible positive and negative edge including negative loops, but no positive loops. Its edges correspond to the roots of the root system "C"_"n"; the column of an edge in the incidence matrix (see below) is the vector representing the root.

* The complete signed graph with half-edges, ±"K"_"n"', is ±"K"_"n" with a half-edge at every vertex. Its edges correspond to the roots of the root system "B"_"n", half-edges corresponding to the unit basis vectors.

* The complete signed link graph, ±"K"_"n", is the same but without loops. Its edges correspond to the roots of the root system "D"_"n".

* An all-positive signed graph has only positive edges. If the underlying graph is "G", the all-positive signing is written +"G".

* An all-negative signed graph has only negative edges. It is balanced if and only if it is bipartite because a circle is positive if and only if it has even length. An all-negative graph with underlying graph "G" is written −"G".

* A signed complete graph has as underlying graph "G" the ordinary complete graph "K"_"n". It may have any signs. Signed complete graphs are equivalent to two-graphs, which are of value in finite group theory. A two-graph can be defined as the class of vertex sets of negative triangles in a signed complete graph.

Adjacency matrix

The adjacency matrix of a signed graph Σ on "n" vertices is an "n" × "n" matrix "A"(Σ). It has a row and column for each vertex. The entry "a"_"vw" in row "v" and column "w" is the number of positive "vw" edges minus the number of negative "vw" edges. On the diagonal, "a"_"vv" = 0 if there are no loops or half-edges; the correct definition when such edges exist depends on the circumstances.

Orientation

A signed graph is oriented when each end of each edge is given a direction, so that in a positive edge the ends are both directed from one endpoint to the other, and in a negative edge either both ends are directed outward, to their own vertices, or both are directed inward, away from their vertices. Thus, an oriented signed graph is the same as a bidirected graph.

Incidence matrix

The (more correctly, "an") incidence matrix of a signed graph with "n" vertices and "m" edges is an "n" × "m" matrix, with a row for each vertex and a column for each edge. It is obtained by orienting the signed graph in any way. Then its entry η_"ij" is +1 if edge "j" is oriented into vertex "i", −1 if edge "j" is oriented out of vertex "i", and 0 if vertex "i" and edge "j" are not incident. This rule applies to a link, whose column will have two nonzero entries with absolute value 1, a half-edge, whose column has a single nonzero entry +1 or −1, and a loose edge, whose column has only zeroes. The column of a loop, however, is all zero if the loop is positive, and if the loop is negative it has entry ±2 in the row corresponding to its incident vertex.

Any two incidence matrices are related by negating some subset of the columns. Thus, for most purposes it makes no difference which orientation we use to define the incidence matrix, and we may speak of the incidence matrix of Σ without worrying about exactly which one it is.

Negating a row of the incidence matrix corresponds to switching the corresponding vertex.

witching

Switching a vertex in Σ means negating the signs of all the edges incident to that vertex. Switching a set of vertices means negating all the edges that have one end in that set and one end in the complementary set. Switching a series of vertices, once each, is the same as switching the whole set at once.

Switching of signed graphs (signed switching) is generalized from Seidel (1976), where it was applied to graphs ("graph switching"), in a way that is equivalent to switching of signed complete graphs.

Switching equivalence means that two graphs are related by switching, and an equivalence class of signed graphs under switching is called a switching class. Sometimes these terms are applied to equivalence of signed graphs under the combination of switching and isomorphism, especially when the graphs are unlabeled; but to distinguish the two concepts the combined equivalence may be called switching isomorphism and an equivalence class under switching isomorphism may be called a switching isomorphism class.

Switching a set of vertices affects the adjacency matrix by negating the rows and columns of the switched vertices. It affects the incidence matrix by negating the rows of the switched vertices.

Fundamental theorem

A signed graph is balanced if and only if its vertex set can be divided into two sets (either of which may be empty), "X" and "Y", so that each edge between the sets is negative and each edge within a set is positive. This is the first theorem of signed graphs (Harary, 1953). It generalizes the theorem that an ordinary (unsigned) graph is bipartite if and only if every cycle has even length.

A simple proof uses the method of switching. To prove Harary's theorem, one shows by induction that Σ can be switched to be all positive if and only if it is balanced.

Frustration

Give each vertex a value of +1 or −1; we call this a state of Σ. An edge is called satisfied if it is positive and both endpoints have the same value, or it is negative and the endpoints have opposite values. The smallest number of frustrated edges in any state is called the frustration index (or line index of balance) of Σ. Finding the frustration index is hard, in fact, it is NP-hard. One can see this by observing that frustration index of an all-negative signed graph is equivalent to the maximum cut problem in graph theory, which is NP-hard.The reason is that the frustration index equals the smallest number of edges whose negation (or, equivalently, deletion; a theorem of Harary) makes Σ balanced. (This can be proved easily by switching.)

The frustration index is important in a model of spin glasses, the mixed Ising model. In this model, the signed graph is fixed. A state consists of giving a "spin", either "up" or "down", to each vertex. We think of spin up as +1 and spin down as −1. Thus, each state has a number of frustrated edges. The energy of a state is larger when it has more frustrated edges, so a ground state is a state with the fewest frustrated energy. Thus, to find the ground state energy of Σ one has to find the frustration index.

Matroid theory

There are two matroids associated with a signed graph, called the "signed-graphic matroid" (also called the "frame matroid" or sometimes "bias matroid") and the "lift matroid", both of which generalize the cycle matroid of a graph. They are special cases of the same matroids of a biased graph.

The signed-graphic matroid "M"("G") (Zaslavsky, 1982) has for its ground set the edge set "E". An edge set is independent if each component contains either no circles or just one circle, which is negative. (In matroid theory a half-edge acts exactly like a negative loop.) A circuit of the matroid is either a positive circle, or a pair of negative circles together with a connecting simple path, such that the two circles are either disjoint (then the connecting path has one end in common with each circle and is otherwise disjoint from both) or share just a single common vertex (in this case the connecting path is that single vertex). The rank of an edge set "S" is "n" − "b", where "n" is the number of vertices of "G" and "b" is the number of balanced components of "S", counting isolated vertices as balanced components. This matroid is the column matroid of the incidence matrix of the signed graph.That is why it describes the linear dependencies of the roots of a classical root system.

The extended lift matroid "L"₀("G") has for its ground set the set "E"₀ the union of edge set "E" with an extra point, which we denote "e"₀. The lift matroid "L"("G") is the extended lift matroid restricted to "E". The extra point acts exactly like a negative loop, so we describe only the lift matroid. An edge set is independent if it contains either no circles or just one circle, which is negative. (This is the same rule that is applied separately to each component in the signed-graphic matroid.) A matroid circuit is either a positive circle or a pair of negative circles that are either disjoint or have just a common vertex. The rank of an edge set "S" is "n" − "c" + ε, where "c" is the number of components of "S", counting isolated vertices, and ε is 0 if "S" is balanced and 1 if it is not.

Other kinds of "signed graph"

Sometimes the signs are taken to be +1 and −1. This is only a difference of notation, if the signs are still multiplied around a circle and the sign of the product is the important thing. However, there are two other ways of treating the edge labels that do not fit into signed graph theory.

The term "signed graph" is applied occasionally to graphs in which each edge has a weight, "w"("e") = +1 or −1. These are not signed graphs; they are weighted graphs with a restricted weight set. The difference is that weights are added, not multiplied. The problems and methods are completely different.

The name is also applied to graphs in which the signs function as colors on the edges. The significance of the color is that it determines various weights applied to the edge, and not that its sign is intrinsically significant. This is the case in knot theory, where the only significance of the signs is that they can be interchanged by the two-element group, but there is no intrinsic difference between positive and negative. The matroid of a sign-colored graph is the cycle matroid of the underlying graph; it is not a frame or lift matroid. The sign labels, instead of changing the matroid, become signs on the elements of the matroid.

In this article we discuss only signed graph theory in the strict sense. For sign-colored graphs see colored matroids.

Generalizations

A signed graph is a special kind of gain graph. The pair ("G", "B"("G")) is a special kind of biased graph.

References

D. Cartwright and F. Harary (1956). Structural balance: a generalization of Heider'stheory. "Psychological Review" 63 (1956), 277-293.

F. Harary (1953). On the notion of balance of a signed graph. "Michigan Mathematical Journal", 2 (1953-1954), 143-146 and addendum preceding page 1.

J.J. Seidel (1976). A survey of two-graphs. In "Colloquio Internazionale sulle Teorie Combinatorie" (Rome, 1973), Tomo I, pages 481-511. Atti dei Convegni Lincei, No. 17. Accademia Nazionale dei Lincei, Rome.

T. Zaslavsky (1982). Signed graphs. "Discrete Applied Mathematics", 4, 47-74. Erratum. "Discrete Applied Mathematics", 5 (1983), 248.