Matrices in the Theory of
[10pt] Signed Simple Graphs
[15pt] Thomas Zaslavsky
Department of Mathematical Sciences
Binghamton University (SUNY)
Binghamton, NY 13902-6000, U.S.A.
[15pt] July 29, 2019
Abstract. I discuss the work of many authors on various matrices used to study signed graphs, concentrating on adjacency and incidence matrices and the closely related topics of Kirchhoff (‘Laplacian’) matrices, line graphs, and very strong regularity.
This article is a survey of the uses of matrices in the theory of simple graphs with signed edges. A great many authors have contributed ideas and results to this field; but amongst them all I have felt to be exceptionally inspiring and important the relevant works of J.J. Seidel and G.R. Vijayakumar.
A signed simple graph is a graph, without loops or parallel edges, in which every edge has been declared positive or negative. Such a signed graph is illustrated in Figure I.1(a). For many purposes the most significant thing about a signed graph is not the actual edge signs, but the sign of each circle (or ‘cycle’ or ‘circuit’), which is the product of the signs of its edges. This fact is manifested in simple operations on the matrices I will present.
I treat three kinds of matrix of a signed graph, all of them direct generalisations of familiar matrices from ordinary, unsigned graph theory.
The first is the adjacency matrix. The adjacency matrix of an ordinary graph has for adjacent vertices; that of a signed graph has or , depending on the sign of the connecting edge. The adjacency matrix leads to questions about eigenvalues and strong regularity.
The second matrix is the vertex-edge incidence matrix. There are two kinds of incidence matrix of an unsigned graph. The unoriented incidence matrix has two ’s in each column, corresponding to the endpoints of the edge whose column it is. The oriented incidence matrix has a and a in each column. For a signed graph, there are both kinds of column, the former corresponding to a negative edge and the latter to a positive edge.
Finally, there is the Kirchhoff matrix.111The Kirchhoff matrix is sometimes called the ‘Laplacian’, but other matrices are also called ‘Laplacian’. I adopt the unambiguous name. This is the adjacency matrix with signs reversed, and with the degrees of the vertices inserted in the diagonal. The Kirchhoff matrix equals the incidence matrix times its transpose. If we multiply in the other order, the transpose times the incidence matrix, we get the adjacency matrix of the line graph, but with ’s in the diagonal.
All this generalises ordinary graph theory. Indeed, much of graph theory generalises to signed graphs, while much—though certainly not all—signed graph theory consists of generalising facts about unsigned graphs.
As this article is expository I will give only elementary proofs, to illustrate the ideas.
I. Fundamentals of Signed Graphs
i.a. Definitions about Vectors and Matrices
A vector of all ’s is denoted by . The matrix consists of all ’s.
i.b. Definitions about Graphs
A graph is with vertex set and edge set . All graphs will be undirected, finite, and simple—without loops or multiple edges—except where explicitly stated otherwise (though most of what I say works well when loops and multiple edges are allowed). The order of the graph is . is the number of connected components of . is the complement of .
An edge with endpoints may be written or . An edge with endpoints may also be written .
The degree of a vertex in a graph is the number of edges incident with . If every vertex has the same degree , we say is regular of degree , or briefly, -regular.
A walk in is a sequence of edges, where the second endpoint of is the first endpoint of . Its length is . Vertices and edges may be repeated. A path is a walk with no repeated vertices or edges. A closed path is a walk of positive length in which , but having no other repeated vertices or edges.
A subgraph of is spanning if it contains all the vertices of . A circle (or circuit, cycle, polygon) is the graph of a closed path; that is, it is a -regular connected subgraph. A theta graph consists of three internally disjoint paths joining two vertices. A pseudoforest is a graph in which every component is a tree or a 1-tree (a tree with one extra edge forming a circle). A block of is a maximal 2-connected subgraph, or an isthmus or an isolated vertex. A cutset is the set of edges between a vertex subset and its complement, except that the empty edge set is not considered a cutset.
The adjacency matrix of is the matrix in which if is an edge and if not. The Seidel adjacency matrix of is the matrix in which if and otherwise is if is an edge, if it is not. Seidel introduced this matrix in ; his very successful use of it in many papers led to his name’s becoming firmly attached. I will have much to say about the Seidel matrix in terms of signed graphs.
i.c. Definitions about Signed Graphs
A signed graph is a pair where is a graph, called the underlying graph, and is the sign function or signature. The sign group can also be written ; I shall treat the two notations as equivalent.222The important thing is that the signs form a multiplicative group of two elements that acts on numbers. For matrix theory the notation is simpler if they are themselves numbers. Often, we write to mean that the underlying graph is . and are the sets of positive and negative edges. The positive and negative subgraphs,
are unsigned graphs.
A signed graph is simply signed if it has no loops333Usually, negative loops would be allowed; but that is not so suitable to the matrix theory. and no parallel edges with the same sign; but it may have two edges, one positive and one negative, joining a pair of vertices. We shall be concerned mostly with signed simple graphs (those with no parallel edges) but occasionally simply signed graphs, and even loops, will play a role.
Signed graphs and are isomorphic if there is a graph isomorphism that preserves edge signs.
Define to be regular if both and are regular graphs.
We say is homogeneous if all edges have the same sign, and heterogeneous otherwise. (I got this handy terminology from M. Acharya and co-authors.) It is all positive or all negative if all edges are positive or negative, respectively. denotes with all positive signs. is with all negative signs.
denotes a complete graph with vertex set , whose edges are negative if they belong to and positive otherwise. That is, and .
i.e. Walks, Circles, and their Signs
The sign of a walk is the product of its edge signs:
Thus, a walk is either positive or negative, depending on whether it has an even or odd number of negative edges, counted with their multiplicity in if there are repeated edges.
A circle, being the graph of a closed path, has a definite sign, either positive or negative. The class of positive, or negative, circles in is denoted by , or , respectively. One can characterise the class of negative circles by the following property. A theta graph is the union of three paths joining the same two vertices, but otherwise disjoint.
Proposition I.1 (cf. ).
A class of circles in a graph is the class of negative circles of a signed graph if and only if every theta subgraph contains an even number of circles in .
A negative circle of length —which consists of one positive edge and one negative edge and can only occur in a signed multigraph—is called a negative digon.
A signed graph , or a subgraph or edge set, is called balanced if every circle in it is positive. is the number of connected components of that are balanced. For , is the number of balanced components of .
A circle is balanced iff it is positive. A walk is called balanced when its underlying graph is balanced; thus, a positive walk may be balanced or unbalanced, and the same holds for a negative walk.
It is easy to see that is balanced if and only if every block is balanced.
Theorem I.2 (Harary’s Balance Theorem ).
A necessary and sufficient condition for to be balanced is that there be a bipartition of into and such that an edge is negative precisely when it has one endpoint in and one in . ( or may be empty.)
In other words, is balanced just when is empty or a cutset. When is balanced, a Harary bipartition is any bipartition as in the theorem. It is unique if and only if is connected. It was not so easy to prove this theorem at the time, but with switching it becomes simple; see later.
is balanced iff is bipartite.
This means balance is a kind of generalisation of biparticity. That turns out to be a valid insight in a number of ways; e.g., in regard to the unoriented incidence matrix of a graph (Example IV.F.2).
We call antibalanced if is balanced; equivalently, if all even circles are positive and all odd circles are negative. The quick way to get an antibalanced signed graph is to give negative signs to all edges of a graph. Corresponding to Harary’s Balance Theorem is the following result:
A necessary and sufficient condition for to be antibalanced is that there be a bipartition of into and such that an edge is positive precisely when it has one endpoint in and one in . ( or may be empty.)
Thus, is antibalanced iff is empty or a cutset. Correspondingly to Corollary I.3, is antibalanced iff is bipartite.
Switching means reversing the signs of all edges between a vertex set and its complement. may be empty. We say is switched in . The switched graph is written . Vertex switching means switching a single vertex. Switching a set has the same effect as switching all the vertices in , one after another.
Another version of switching, which is equivalent to the preceding and is very useful, is in terms of a function , called a switching function. Switching by means changing to defined by
The switched graph is written
If can be switched to become , we say and are switching equivalent. Switching equivalence is an equivalence relation on signatures of a fixed graph. An equivalence class is called a switching class. denotes the switching class of .
If is isomorphic to a switching of , we say and are switching isomorphic. Switching isomorphism is an equivalence relation on all signed graphs. Often in the literature switching isomorphism is not distinguished from switching equivalence, but I prefer to separate the two concepts.
Lemma I.5 (Switching Lemma [35, Corollary 3.3]).
is balanced if and only if it switches to an all-positive signature, and it is antibalanced if and only if it switches to an all-negative signature.
To prove the first statement one can assume is connected. Take a spanning tree, rooted at any vertex , and switch so the tree is all positive. The switching function for this is , where is the unique -path in . is balanced if and only if there are no remaining negative edges.
The second statement follows by a similar proof, or by negation from the first part. ∎
A useful way to think of the Switching Lemma without actually switching is that is balanced iff there is a function such that . This amounts to saying that has a potential function, i.e., .
Properties preserved by switching are the signs of circles, and balance or imbalance of and of any subgraph; also deletion sets and negation sets.
The proof technique of the Switching Lemma yields a valuable insight into equivalence of signed graphs.
Two signed graphs with the same underlying graph are switching equivalent if and only if they have the same class of positive circles.
As in the previous proof we may assume the graphs, and , are connected. Switch both signed graphs so a fixed spanning tree is all positive. Call the switched graphs and .
If and are switching equivalent, so are and , but as they agree on a spanning tree, one can only be switched to the other by no switching at all or by switching every vertex. Thus, , whence .
If , then after switching . Now, with an all-positive spanning tree , the only possible difference between and is in the signs of the non-tree edges. But the sign of an edge is the sign of the unique circle in , which is the same in and . Therefore, . As and switch to the same signed graph , they are switching equivalent. ∎
Corollary I.7 (Switching Isomorphism ).
Two signed graphs are switching isomorphic if and only if there is an isomorphism of underlying graphs that preserves the signs of circles.
With switching I can give short proofs of such results as Harary’s fundamental theorem.
Proof of Harary’s Balance Theorem.
If there is such a bipartition, every circle has an even number of negative edges, so is balanced.
If is balanced, switch it to be all positive. Letting be the set of switched vertices, the bipartition is . ∎
Signed graphs and balance were invented by Harary by 1953  to treat a question in social psychology . They have since been reinvented over and over in many contexts in physics, geometry, economics, and more—thus showing that they are a natural concept—but it was Harary who first had the idea of labelling with the 2-element group by putting signs on the edges and multiplying them. Harary also introduced antibalance, in .
It is remarkable that, years before Harary, König [19, Section X.3] had the idea of a graph with a distinguished subset of edges, in a way we now recognise as equivalent to signed graphs, and even proved Harary’s Balance Theorem and defined switching in the form of taking the set sum of a cutset with the set of negative edges. Despite all this, he failed to notice the value of edge labels that one can multiply, which I regard as the crucial step in the invention of signed graphs.
The first to think of switching as an operation on signed graphs were the social psychologists Abelson and Rosenberg . However, their formalism for switching, in terms of a Hadamard product operation on their adjacency matrix (see Section II.F), was awkward and ungraphical. I arrived at switching of signed graphs as an adaptation and generalisation of graph switching, introduced by Seidel and applied with great effect in many papers (cf. [20, 25]). Switching, like signed graphs themselves, has been reinvented several times, with varying names and notation.
The observation following the proof of the Switching Lemma was published, independently of other work on switching, by Sampathkumar .
Ii. Adjacency Matrices
The adjacency matrix is an matrix in which (the sign of the edge ) if and are adjacent, and if they are not. Thus is a symmetric matrix with entries and zero diagonal, and conversely, any such matrix is the adjacency matrix of a signed simple graph. The absolute value matrix, , equals , the adjacency matrix of the underlying unsigned graph.
When has multiple edges, the entry of is the sum of the signs of all edges. Positive and negative edges cancel each other. This will become important to us in the treatment of line graphs.
ii.b. Walks and Neighbors
Powers of count walks in a signed way. Let be the number of positive walks of length from to (that is, the sign product of the edges in is positive), and let be the number of negative walks.
The entry of is .
I omit the proof, which is not difficult.
A noteworthy special case is the square of . Let denote the number of common positive neighbors of distinct vertices and , the number of their common negative neighbors, and the number of neighbors that are positive neighbors of one and negative neighbors of the other.
In the entry is and the entry, for , is
The diagonal entry is the number of walks , since every such walk has the positive sign . This number equals the number of neighbors of . As for the off-diagonal entry, and . ∎
If is complete, i.e., , then is the Seidel adjacency matrix of the negative subgraph . That means the Seidel matrix of a graph is the adjacency matrix of a signed complete graph. This fact inspired my work on adjacency matrices of signed graphs.
If is bipartite, so that , then where is an matrix of ’s, ’s, and ’s. If is complete bipartite, has no ’s.
Switching has a simple effect on . Given , switching by negates both the rows and columns of vertices in . Given a function , switching by negates both the rows and columns of vertices with . In matrix terms, let be the diagonal -matrix with in the th diagonal position. Then switching by conjugates by ; that is,
(Note that ; I inserted the inversion in order to show that this is truly conjugation.) We may conclude that:
Signed graphs and on the same vertex set are switching equivalent if and only if their adjacency matrices satisfy for some diagonal -matrix whose diagonal has no zeros.
We can strengthen this to a criterion for switching isomorphism (if not one that is computationally practical). By ‘similarly rearranging’ the rows and columns of a square matrix , I mean that one applies the same permutation to the rows and to the columns.
Signed graphs and are switching isomorphic if and only if the rows and columns of can be similarly rearranged so that for some diagonal -matrix whose diagonal has no zeros.
I omit the simple verifications.
Similarly rearranging the rows and columns is represented with matrices as conjugation by a permutation matrix , i.e., is changed to . Thus, in strictly matrix language, .
ii.e. Eigenvalues and Eigenvectors
An eigenvalue of is an eigenvalue of its adjacency matrix. The spectrum of is the list of its eigenvalues with their multiplicities. Individual eigenvalues and the spectrum may give information about . I know of nothing that has been done on this aside from special cases: signed graphs whose underlying graph is regular, which can be treated through the Kirchhoff matrix (see Section IV.E); those with no eigenvalues greater than (they are, with a few exceptions, line graphs of signed graphs; see Section V.E); and Acharya’s matricial criterion for balance. There is one fundamental result that must be the starting point for all investigations:
Switching a signed graph does not change its spectrum.
This is a corollary of Proposition II.3, because conjugating a matrix does not change its spectrum. ∎
Proposition II.6 (B.D. Acharya ).
is balanced if and only if has the same eigenvalues (with multiplicities) as does .
I omit the proof. Acharya’s criterion is the more interesting because it cannot be strengthened to an eigenvalue criterion for switching isomorphism. Although switching-isomorphic signed graphs obviously must have the same eigenvalues, the converse is false, as one can see from the facts that nonisomorphic unsigned graphs can have the same eigenvalues and that .
A regular signed graph has as an eigenvalue, associated to the eigenvector .
is regular if and only if is an eigenvector of both and .
The term of in is , the sum of degrees in the positive and negative subgraphs. The term of in is , the difference of degrees. A necessary and sufficient condition for to be an eigenvector of both is that the sum and difference be independent of ; equivalently, that the positive and negative subgraphs be regular; thus, by definition, that be regular. ∎
Although it does not directly provide information about the signed graph itself, from matrix theory we do know something about the relationship between eigenvalues of and those of induced subgraphs.
If is an induced subgraph of , then the largest eigenvalues satisfy .
This is an instance of the general interlacing theorem for eigenvalues of real, symmetric matrices [9, Theorem 9.1.1]. ∎
McKee and Smyth  start with an eigenvalue question and arrive at signed graphs. If a symmetric integer matrix has all its eigenvalues (which are necessarily real) in the interval , then it is the adjacency matrix of a signed simple graph except for having possibly nonzero elements on the diagonal. The signed graphs that can appear are subgraphs of members of three infinite families and a handful of sporadic examples. Their result is one of several reasons to think that are especially significant bounds on eigenvalues. I refer the reader to their long paper for the details.
As I mentioned, the eigenvalue properties of signed graphs are an open field of inquiry.
ii.f. The Abelson–Rosenberg Adjacency Matrix
An oddity from early in signed graph theory is the adjacency matrix formulated by the inventive social psychologists Abelson and Rosenberg . Their matrix, , has entries (and these symbols have addition and multiplication operations), where stands for nonadjacency, and are for positive and negative adjacency, and is for vertices that are adjacent by both a positive and a negative edge; and it has on the diagonal instead of . Abelson and Rosenberg employed their symbols to stand for an ‘unrelated’, ‘positive’, ‘negative’, and ‘ambivalent’ relationship between the vertices, which represented persons in a social group. The symbol , which can occur with a simply signed graph but not a signed simple graph, was necessary for the psychology.
Harary, Norman, and Cartwright [14, Theorem 13.8] used to show the existence of walks of given length and sign between two specified vertices. Their theorem says there is a positive (resp., negative) walk of length from to iff the position in contains or (resp., or ).
The degree of a vertex has several generalisations to signed graphs. The underlying degree is the degree in the underlying graph, that is, the total number of edges incident with . The positive or negative degree, or , is the degree in the positive subgraph or the negative subgraph . The net degree is
If is regular, we can speak of its degree (of any kind): it is the degree of any vertex. For example, the net degree of is for every vertex . is regular if and only if is regular and for some real number , which necessarily equals
ii.h. Very Strong Regularity
Seidel discovered that a strongly regular graph has a nice definition in terms of its Seidel adjacency matrix , namely, that
for some constants (thus in particular is an eigenvector of ). (See [24, 25].) The constants have combinatorial interpretations. For a start, , the degree of . Given an edge , the number of vertices adjacent to exactly one of and equals . Given a nonadjacent pair , that number is . If , the second defining equation is superfluous.
Contemplating the fact that is the same matrix as , I was somehow led to the following definition: A signed graph is very strongly regular  if its adjacency matrix satisfies
for some constants . Here is the adjacency matrix of the complement of . The combinatorial interpretation of these parameters is:
, the net degree of every vertex (hence it is an integer). Hence is regular.
where are the numbers of positive and negative triangles on an edge .
for any pair of nonadjacent vertices, where are the numbers of positive and negative length-2 paths joining the vertices.
and are independent of the choices of adjacent or nonadjacent vertices.
The big problem is to, in some sense, classify very strongly regular signed graphs. I am currently working on this. Various simplifications are helpful. For instance, , whose signature is , behaves just like except that and are negated. Thus, one may assume that is nonnegative, or that is not all negative, when it is convenient to do so.
The most important factor in the classification is which of and are 0. As is usual in such problems, there are strong numerical restrictions on the values of . It is interesting that some of the types include kinds of matrices that have already been studied for many years. I will run down the possibilities. I write for . A weighing matrix is a -matrix such that is a multiple of .
Homogeneous signed graphs can be assumed (by negation) to be all positive. is very strongly regular if and only if is a strongly regular unsigned graph. This demonstrates that we have a true generalisation of strong regularity.
For the rest of the cases I assume is inhomogeneous.
When , the defining equations are and . The eigenvalues are , one of which is . Consequently, is a square and is a symmetric weighing matrix with zero diagonal and in each row entries equal to and entries equal to .
When but , the defining equations are and . Solving for the eigenvalues shows there is an integer such that
When , from the eigenvector we deduce that which significantly constrains the numbers. This case is too complicated for further description here. For instance, the eigenvalues of depend on those of .
Much more can be said, but not here; see .
To my knowledge, the first adjacency matrix of a signed graph was that of Abelson and Rosenberg. The standard adjacency matrix, , appeared soon after, but I am not sure exactly when and where. Harary certainly used it early on.
Switching of the standard adjacency matrix is implicit in  and explicit in a form that is equivalent to switching signed complete graphs in  but was not made explicitly and generally signed-graphic until later (I am not sure just when). Abelson and Rosenberg, as I mentioned, had the concept of switching but they did not develop it far, and I think that would have been difficult given their mathematical formulation.
iii.a. Bidirected Graphs
In a bidirected graph, every edge has an independent orientation at each end. (This concept is due to Jack Edmonds; cf. .) We think of these in two ways: as an arrow at each end, which may point towards or away from the endpoint, and as a sign on the end of that is incident with , which is if the arrow points to the endpoint, if the arrow is directed away from the endpoint.
iii.b. Oriented Signed Graphs
A bidirected graph is naturally signed by the formula
for an edge . An edge is negative if its arrows both point toward their corresponding endpoints (I call such edges extraverted), or both away from their endpoints (I call these introverted edges). An edge is positive if one arrow points at its endpoint while the other is directed away from its endpoint. Thus, a positive edge is just like an ordinary directed edge.
Conversely, an orientation of a signed graph is a bidirection of that satisfies the sign formula (III.1).
Reorienting an edge means replacing and by their negatives. In terms of arrows, it means reversing the arrows at both endpoints of the edge. This does not change the sign of the edge.
One switches a vertex set in a bidirected graph by changing the signs of all the edge ends incident with a vertex in . In terms of a switching function , one has .
Recall that switching the entire vertex set does not change a signed graph; but it does have an effect on an orientation: it reverses all the arrows.
Iv. Incidence Matrices
An incidence matrix of is a matrix in which the column of edge has two entries , one in the row of each endpoint of , and 0’s elsewhere. The two nonzero entries must have product equal to ; that is, they are equal if is negative, but if is positive, one is and the other is .
The notation I use for an incidence matrix of a signed graph is (read ‘Eta’)
The incidence matrix is not unique. The choice of signs in each column reflects a choice of orientation of ; in fact, the entry in is equal to
(Thus, we may define in the latter case. Then ; the only difference between the two is the point of view: the former is regarded as a bidirection and the latter is a matrix entry.) Conversely, the entries of an incidence matrix of determine an orientation .
The incidence matrix can be treated as a matrix over any field , or indeed any ring, in particular over the ring of integers, or the -element field.
iv.b. Switching and Reorientation
The effect on of switching is to negate the rows corresponding to the vertices in . In terms of a switching function and the matrix , switches to .
The effect of reorienting edges is to negate the corresponding columns of .
The rank of is a basic fact that generalises more widely known but limited results about incidence matrices.
If has characteristic , then has rank . Otherwise, has rank .
As can be written in the block form
where are the connected components of , it suffices to prove the rank formula for a connected signed graph. For , write for the submatrix of that consists of the columns corresponding to .
Let be a spanning tree of and switch so is all positive. If is balanced, then after switching it is all positive. Then its incidence matrix is identical with that of , which is well known to have rank (not , because the rows sum to ; not less than , because after dropping one row has determinant ).
If in , then the definition shows that over , and by the preceding argument the rank is .
If is unbalanced, there is a negative edge . Consider the submatrix of that consists of the columns corresponding to and . It has the form The last column has two ’s or two ’s, so its sum is . The only linear dependence among the rows of is that they sum to ; but the sum of the rows of the matrix has in its last column. Therefore, if in the rows of are linearly independent; this submatrix has rank . Thus, has rank . ∎
Since the incidence matrix of has rank , and when, and only when, is balanced, has the same rank as if and only if is balanced.
iv.d. The Kirchhoff Matrix and Matrix-Tree Theorems
The Kirchhoff matrix (also called the Laplacian) is
where is the degree matrix of , that is, the diagonal matrix whose diagonal entries are the degrees of the vertices in the underlying graph.
The rank of (over the real numbers) is . Its nullity is .
A matrix of the form is positive semidefinite and has rank equal to the rank of . By standard matrix theory, its eigenvalues are all non-negative. In our case, has rank by Theorem IV.1. ∎
The classical Matrix-Tree Theorem expresses the number of spanning trees of a graph in terms of the Kirchhoff matrix: the number is where is with any one row and column deleted. I state two signed-graphic analogs. For proofs see the references; the first is not hard (it uses the Binet–Cauchy theorem in the standard way), but the second is rather complicated both to state fully and to prove.
Theorem IV.3 (Zaslavsky [35, Section 8A]).
The determinant is the sum, over all pseudoforests with edges and with no positive circles, of , where is the number of components of .
The reason is that . Note that the pseudoforests in the theorem, because they have as many edges as vertices, can have no tree components.
Theorem IV.4 (Chaiken’s All-Minors Matrix-Tree Theorem ).
From delete rows, corresponding to vertices , and columns, corresponding to , and then take the determinant. The resulting number is the sum of , where is the number of circles in , over all -edge spanning pseudoforests without positive circles such that each tree component of contains exactly one and one . The sign of the term is given by a complicated rule.
Chaiken’s theorem is even more general: the edges can be weighted.
iv.e. Eigenvalues of the Kirchhoff Matrix
The eigenvalues of the Kirchhoff matrix are a natural topic of study. As a simple example, they constrain the eigenvalues of when the underlying graph is regular, in just the same way as with ordinary graphs.
If the underlying graph of is regular of degree , then all eigenvalues of are . The value is an eigenvalue of multiplicity ; thus, is an eigenvalue if and only if has a balanced component.
If is regular of degree , then . Thus, . The Kirchhoff matrix is positive semidefinite and has nullity by Lemma IV.2. Therefore, all eigenvalues of are at least . The eigenvalues of are determined by where are the eigenvalues of . It follows that all and that the eigenvalue of has the same multiplicity as the corresponding eigenvalue of . The multiplicity of as an eigenvalue of is the nullity of . That gives the multiplicity of as an eigenvalue of . ∎
Hou, Li, and Pan  studied the eigenvalues of in general, without assuming regularity of the underlying graph. They have two kinds of results: upper and lower bounds on the largest eigenvalue of (mostly in terms of the underlying graph), and an interlacing theorem for all eigenvalues. Write the eigenvalues in decreasing order as . Here are some of their results:
Theorem IV.6 (Hou, Li, and Pan ).
Let be a connected simple graph and let be a signed simple graph.
(1) , with equality iff is antibalanced (Lemma 3.1).
(2) , with equality iff switches to (Theorem 3.4).
(3) (Corollary 3.8).
(4) (Theorem 3.10).
Theorem IV.7 (Hou, Li, and Pan [16, Lemma 3.7]).
An oriented incidence matrix of the unsigned graph is the same as an incidence matrix of the all-positive signed graph . Since is balanced, the rank given by our formula equals , as is well known.
The unoriented incidence matrix of is a -matrix which has in position if is an endpoint of and has otherwise. (This matrix is often, though I think unfortunately, called simply ‘the incidence matrix’ of .) It is an incidence matrix of the all-negative signed graph, . Since an all-negative graph is balanced if and only if it is bipartite (Corollary I.3), the rank of the matrix (except in characteristic ) equals where is the number of bipartite components of . This result was previously obtained by ad hoc methods (originally by van Nuffelen ), but it is really a special case of the general rank theorem for signed graphs.
The Kirchhoff matrix equals . Its determinant equals the sum of over all -edge pseudoforests of which no component is bipartite.
The signed graphs of Figure I.1 are connected and unbalanced. Therefore, each of their incidence matrices has rank , which is .
V. Line Graphs
v.a. Unsigned Line Graphs
The line graph of an unsigned graph is denoted by . Its vertex set is , and two edges are adjacent if they have a common endpoint in . has two kinds of distinguished circles: vertex triangles are formed by three edges incident with a common vertex, and derived circles are the line graphs of circles in . Every circle in is known to be a set sum (i.e., symmetric difference of edge sets) of vertex triangles and derived circles.
v.b. Signed Line Graphs
The line graph of is a switching class, not a single signed graph . Its underlying graph is the line graph of the underlying graph. To define we may take the approach of edge orientation or a direct definition of the circle signs.
Definition by Orientation .
Choose an orientation of . We define a bidirection of and therefore an edge signature, thus forming the line signed graph . Two -edges incident with a vertex form an edge in , whose vertices are and . An end of may therefore be written , corresponding to the end in . Define .
In terms of arrows, bidirect each edge of with two arrows as indicated by , and let the arrow on point into the vertex iff the arrow on points into the vertex in .
Reorienting an edge in corresponds to switching the corresponding vertex in . Thus, is well defined only up to switching. I.e., it is a well defined switching class.
Make every vertex triangle negative and give to every derived circle the same sign as the circle in it derives from. Other circles get signed by the following sum rule: If is the set sum of certain vertex triangles and derived circles, its sign is the product of the signs of those vertex triangles and derived circles. This rule is a consequence of Proposition I.1. One has to prove that the sum rule gives the same sign no matter how it is applied, which is most easily achieved by showing that the definition by circle signs agrees with that by edge orientation, the latter being obviously well defined.
v.c. Reduced Line Graphs
If we allow to have parallel edges—though only with opposite sign, so that is simply signed but two vertices can be joined by both a positive and a negative edge—there is a definition of line graph similar to the preceding one. In the line graph there are negative digons derived from those of . In the line-graph adjacency matrix , these digon edges cancel. To represent that phenomenon accurately in the line graph we should reduce the line graph by deleting pairs of parallel edges, and , of opposite sign. The reduced line graph, , is what results. It is a signed simple graph.
The importance of reduction is that, even though the underlying graph of itself may not be simple, that of is. More precisely, if the underlying graph is simple, is already simple and does not need to be reduced. If is not simple, then is not simple but the underlying graph of the reduced line graph is simple. Thus, amongst the signed simple graphs, there are graphs that are reduced line graphs of simply signed graphs but not line graphs of simply signed graphs. One is shown in Figure V.2; since is not a line graph, cannot be an unreduced line graph.
This larger class, the reduced line graphs, has been studied by Vijayakumar and his coworkers, under the name of ‘signed graphs represented by ’ [30, 6]. From our line-graphic point of view, the name comes from the fact that
where is the standard orthonormal basis of , is the set of all those vectors that can be the column of an edge in the incidence matrix of a signed graph with vertices; the line graph corresponds to taking dot products of these vectors. Allowing to be arbitrarily large, one has .
is best known as one of the classical root systems in Lie algebra. The other important root system for us is , an exceptional root system in whose exact definition is unnecessary here—but see [31, 32].
We need to define representation by since it depends on the adjacency matrix and is at the heart of the treatment of line graphs. A signed simple graph is represented by if there is an injection such that the dot product if and are adjacent and if they are not, and all . (That ought to have been to be consistent with our definition of line graph but the negative sign was omitted by Vijayakumar, who defined this terminology.)
v.e. Adjacency Matrix and Eigenvalues
The adjacency matrix of is the matrix given by
The largest eigenvalue of is at most . Moreover, is an eigenvalue of with multiplicity ; in particular, it is an eigenvalue if and only if has a connected component that is neither a tree nor an unbalanced 1-tree.
A matrix product of the form is positive semidefinite, so all its eigenvalues are at least . An eigenvalue of corresponds to the eigenvalue of . Letting and , we deduce that the eigenvalues of the right side of Equation (V.1) are not greater than .
By matrix theory, the rank of equals that of , which is . It has as an eigenvalue of multiplicity (order rank). The order of is . Thus, it has as an eigenvalue of multiplicity . The corresponding eigenvalue of has the same multiplicity. ∎
If is a signed simple graph whose eigenvalues are , then is represented by or .
That is, the connected signed simple graphs whose eigenvalues are at most are the reduced line graphs of simply signed graphs (represented by ) and only a few additional sporadic examples (represented by ) which are neither line graphs nor reduced line graphs. This fact was explicitly recognised by Chawathe and Vijayakumar in [6, Theorem 1.1] (or see [33, the first Theorem 2.4, on p. 214]), although the proof is actually in the classic paper of Cameron, Goethals, Seidel, and Shult .
The essential observation behind Theorem V.2 is that, if has all eigenvalues , then is a positive semidefinite matrix and therefore is the matrix of inner products of a set of vectors in for some . The proof involves classifying the possible sets , which turn out to be the subsets of and .
Singhi and Vijayakumar showed that having an eigenvalue implies an induced subgraph with all eigenvalues ; indeed, their result is stronger:
Theorem V.3 (Singhi and Vijayakumar ).
If a signed simple graph has an eigenvalue greater than or equal to , then it contains an induced subgraph whose largest eigenvalue is exactly .
This result is the harder converse of the following corollary of Proposition II.8.
If has an induced subgraph whose largest eigenvalue is , then the largest eigenvalue of is at least .
The line graph of the all-negative signed graph is , the switching class of the ordinary line graph with all negative signs. From the standpoint of line graphs, therefore, ordinary graphs should be considered as all-negative signed graphs, instead of all-positive as in other parts of signed graph theory.
The line graphs that are antibalanced are those of the form where is an ordinary line graph or a generalised line graph; that is:
Theorem V.5 (Cameron, Goethals, Seidel, and Shult ).
The all-negative signed graphs that are reduced line graphs of signed graphs are precisely the all-negative generalised line graphs .
(See the statement in the introduction to  [where ‘The family of sigraphs represented by ’ should be ‘The family of graphs …’].) Again, we see that the usual line graphs are all-negative signed graphs. (This result was not originally stated explicitly in terms of signed graphs, but in an equivalent fashion in terms of adjacency matrices.)
As a directed graph is a bidirected all-positive graph , it has a signed line graph. The positive part is a directed graph as well; it is precisely the Harary–Norman line digraph of .
The two definitions of a line graph of a signed graph are from , which has been on the verge of being written for more than two decades.
The interpretation of graphs represented by , including Hoffman’s generalised line graphs (Example V.F.2), as line graphs was first stated in [36, Example 2]. It is implicit in the geometrical representation in , in Vijayakumar’s geometrical representation by , and also in Godsil and Royle’s presentation [9, Section 12.1], based on the seminal paper , which talks of cancelling double edges but without recognising the cancellation as due to opposite signs.
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