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A **signed** digraph is a directed graph with **signed** arcs. **Signed** digraphs are far more complicated than **signed** graphs, because only the signs of directed cycles are significant. For instance, there are several definitions of balance, each of which is hard to characterize, in strong contrast with the situation for **signed** undirected graphs.

**Signed** digraphs should not be confused with oriented **signed** graphs. The latter are bidirected graphs, not directed graphs (except in the trivial case of all positive signs).

The sum of two finite **signed** measures is a finite **signed** measure, as is the product of a finite **signed** measure by a real number: they are closed under linear combination. It follows that the set of finite **signed** measures on a measurable space (X, Σ) is a real vector space; this is in contrast to positive measures, which are only closed under conical combination, and thus form a convex cone but not a vector space. Furthermore, the total variation defines a norm in respect to which the space of finite **signed** measures becomes a Banach space. This space has even more structure, in that it can be shown to be a Dedekind complete Banach lattice and in so doing the Radon–Nikodym theorem can be shown to be a special case of the Freudenthal spectral theorem.

If X is a compact separable space, then the space of finite **signed** Baire measures is the dual of the real Banach space of all continuous real-valued functions on X, by the Riesz–Markov–Kakutani representation theorem.

The term **signed** graph is applied occasionally to graphs in which each edge has a weight, w(e) = +1 or −1. These are not the same kind of **signed** graph; 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.

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.

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

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 the frame or lift matroid of the **signed** graph. The sign labels, instead of changing the matroid, become signs on the elements of the matroid.

This approach is directly comparable to the common way of showing a sign (placing a "+" or "−" next to the number's magnitude). Some early binary computers (e.g., IBM 7090) use this representation, perhaps because of its natural relation to common usage. **Signed** magnitude is the most common way of representing the significand in floating point values.

This representation is also called "sign–magnitude" or "sign and magnitude" representation. In this approach, a number's sign is represented with a sign bit: setting that bit (often the most significant bit) to 0 for a positive number or positive zero, and setting it to 1 for a negative number or negative zero. The remaining bits in the number indicate the magnitude (or absolute value). For example, in an eight-bit byte, only seven bits represent the magnitude, which can range from 0000000 (0) to 1111111 (127). Thus numbers ranging from −127 10 to +127 10 can be represented once the sign bit (the eighth bit) is added. For example, −43 10 encoded in an eight-bit byte is 1 0101011 while 43 10 is 0 0101011. A consequence of using **signed** magnitude representation is that there are two ways to represent zero, 00000000 (0) and 10000000 (−0).

There are two slightly different concepts of a **signed** measure, depending on whether or not one allows it to take infinite values. In research papers and advanced books **signed** measures are usually only allowed to take finite values, while undergraduate textbooks often allow them to take infinite values. To avoid confusion, this article will call these two cases "finite **signed** measures" and "extended **signed** measures".

The IEEE 754 standard for floating-point arithmetic (presently used by most computers and programming languages that support floating point numbers) requires both +0 and −0. Real arithmetic with **signed** zeros can be considered a variant of the extended real number line such that 1/−0 = −∞ and 1/+0 = +∞; division is only undefined for ±0/±0 and ±∞/±∞.

This **signed** measure takes only finite values. To allow it to take +∞ as a value, one needs to replace the assumption about f being absolutely integrable with the more relaxed condition

**Signed** Japanese (日本語対応手話, Manually Coded Japanese), is a manually coded form of Japanese that uses the signs of Japanese Sign Language. It is not a natural form of communication among deaf people. It is not common, as sign language was banned in schools until 2002, and oral education was used instead. **Signed** Japanese has some similarities to Pidgin **Signed** Japanese, which may be used by non-native signers.

The Hahn decomposition theorem states that given a **signed** measure μ, there exist two measurable sets P and N such that:

**Signed** French (Français Signé) is any of at least three manually coded forms of French that apply the words (signs) of a national sign language to French word order or grammar. In France, **Signed** French uses the signs of French Sign Language; the Belgium system uses the signs of French Belgian Sign Language, and in Canada the signs of Quebec Sign Language. **Signed** French is used in education and for simultaneous translation, not as a natural form of communication among deaf people.

In physics, **signed** graphs are a natural context for the general, nonferromagnetic Ising model, which is applied to the study of spin glasses.

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. An edge that is not satisfied is called frustrated. The smallest number of frustrated edges over all states is called the frustration index (or line index of balance) of Σ. Finding the frustration index is hard, in fact, it is NP-hard. Aref et al. suggest binary programming models that are capable of computing the frustration index of graphs with up to 10 5 edges in a reasonable time. One can see the NP-hard complexity by observing that the frustration index of an all-negative **signed** graph is equivalent to the maximum cut problem in graph theory, which is NP-hard. The reason for the equivalence 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.)

A **signed** overpunch is a code used to store the sign of a number by changing the last digit. It is used in character data on IBM mainframes by languages such as COBOL, PL/I, and RPG. Its purpose is to save a character that would otherwise be used by the sign digit. The code is derived from the Hollerith Punched Card Code, where both a digit and a sign can be entered in the same card column. Character data which may contain overpunches is called zoned decimal.

**Signed** Spanish and **Signed** Exact Spanish are any of several manually coded forms of Spanish that apply the words (signs) of a national sign language to Spanish word order or grammar. In Mexico, **Signed** Spanish uses the signs of Mexican Sign Language; in Spain, it uses the signs of Spanish Sign Language, and there is a parallel **Signed** Catalan that uses the signs of Catalan Sign Language along with oral Catalan. **Signed** Spanish is used in education and for simultaneous translation, not as a natural form of communication among deaf people. The difference between **Signed** Spanish and **Signed** Exact Spanish is that while **Signed** Spanish uses the signs (but not the grammar) of Spanish Sign Language, and augments them with signs for Spanish suffixes such as -dor and -ción, and with fingerspelling for articles and pronouns, **Signed** Exact Spanish (and **Signed** Exact Catalan) has additional signs for the many grammatical inflections of oral Spanish. All **signed** forms of Spanish drop the grammatical inflections of the sign languages they take their vocabulary from.