Graphs, Gain Graphs, and Geometry a.k.a. Signed Graphs and their Friends Course notes for Math 581: Graphs and Geometry Fall, 2008 | Spring{Summer, 2009 | Spring, 2010 Math 581: Signed Graphs Fall, 2014 Binghamton University (SUNY) Version of November 23, 2014 Lectures by Thomas Zaslavsky Notes by Richard Behr, Peter Cohen, Melissa Fuentes, Simon Joyce, Jackie Kaminski, Yash Lodha, Nathan Reff, Ting Su, and Thomas Zaslavsky Redaction by Thomas Zaslavsky Copyright c 2008{2014 Thomas Zaslavsky Contents Chapter II. Signed Graphs 2 A. Introduction to Signed Graphs 2 A.1. What a signed graph is 2 A.2. Examples of signed graphs 3 A.3. Degrees 3 A.4. Walk and circle signs 4 A.5. Balance 4 A.6. Switching 6 Switching and balance 7 Switching equivalence and switching isomorphism 9 B. Characterizing Signed Graphs 10 B.1. Signature as a homomorphism 10 B.2. Balanced circles and theta oddity 11 C. Connection 13 C.1. Balanced components 13 C.2. Unbalanced blocks 15 C.2.1. Menger's theorem 15 C.2.2. Vertices and edges in unbalanced blocks 16 D. Imbalance and its Measurement 16 D.1. Ways to balance 16 D.1.1. Balancing edges 17 D.1.2. Balancing sets 18 D.2. A plethora of measures 19 D.3. Frustration index 20 D.3.1. Properties 20 D.3.2. Computational complexity 22 D.3.3. Computational methods 23 D.4. Maximum frustration 23 D.5. Disjoint negative circles 26 E. Minors of Signed Graphs 27 E.1. Contraction 27 E.1.1. Contracting a single edge. 27 E.1.2. Contracting an edge set S. 27 E.2. Minors 30 F. Closure and Closed Sets 33 F.1. Closure operator 33 Closure via frame circuits 35 F.2. Closed sets 36 F.3. Signed partial partitions 37 F.3.1. Partial partitions 37 F.3.2. Signed partial partitions 38 F.3.3. Signed partitions 39 F.4. Examples of signed graphs and closed sets 40 ii Table of Contents G. Incidence and Adjacency Matrices 43 G.1. Incidence matrix 43 G.2. Incidence matrix and frame circuits 44 G.3. Adjacency matrix 48 H. Orientation 49 H.1. Bidirected graphs 50 H.2. Incidence matrix of a bidirected graph 51 Walks and coherence 51 I. Equations from Edges, and Signed Graphic Hyperplane Arrangements 52 I.1. Equations from edges 52 I.2. Additive representations: binary and affine 53 J. Chromatic Functions 55 J.1. Coloring a signed graph 56 J.2. Chromatic numbers 57 J.3. Chromatic polynomials 57 J.4. Counting acyclic orientations 61 The sesquijection of acyclic orientations 61 Proper and compatible pairs 71 The number of acyclic orientations 71 J.5. The dichromatic and corank-nullity polynomials 72 Dichromatic polynomials 72 Corank-nullity polynomials 74 J.6. Counting colorations 75 K. Signed Complete Graphs 76 K.1. Coloring 76 Open questions on coloring of signed complete graphs 78 K.2. Two-graphs 78 Graph switching 78 The multiplicity trick 80 K.3. Strongly regular graphs 81 The combinatorics of a detached vertex 82 The matrix of a detached vertex 83 From a strongly regular graph 83 To a strongly regular graph 83 L. Line Graphs of Signed Graphs 84 L.1. What are line graphs for? 84 L.2. Ideas for the line graph of a signed graph 85 Two previous definitions 85 The definition through bidirection 86 M. Circuits, Cocircuits, and their Spaces 87 M.1. Unsigned graphs 88 M.1.1. Cycles, cuts, circuits and bonds 88 M.1.2. Directed cycles and cuts. Indicator vectors. 88 M.2. Signed graphs 90 M.3. Indicator vector of a directed walk 91 M.4. Flows and Cycles 93 Table of Contents 1 M.4.1. Flows 93 M.5. Cuts 95 M.6. The three types of cut 97 Two kinds of balancing set 97 Cuts 97 M.7. Spaces and orthogonality 100 Flows and 1-cycles 100 M.7.1. Cuts 102 Cuts and minimal cuts 102 Readings and Bibliography 104 A. Background 104 B. Signed Graphs 105 C. Geometry of Signed Graphs 105 D. Gain Graphs 106 E. Biased Graphs and Gain Graphs 106 F. General References 106 Chapter II. Signed Graphs Oct 6: Now at last we've arrived at the meat of the course.1 Our purpose is to generalize graph Jackie theory to signed graphs. Not all of graph theory does so generalize, but an enormous amount Kaminski of it does|or should, if the effort were made. Since that has not happened yet, there is plenty of room for a fertile imagination to create new graph theory about signed graphs. A. Introduction to Signed Graphs [[LABEL 2.basics]][[LABEL 2.sg]] A signed graph is a graph with signed edges. But what, precisely, does that mean? In fact, not every edge has a sign; it is only ordinary edges|links and loops|that do. A.1. What a signed graph is. [[LABEL 2.sgintro]] We give two definitions. The first is the simpler: every edge gets a sign. The cost is that we cannot have loose or half edges; but as we shall see in the treatment of contraction (Section E.1) that is rather too constraining, whence the second, more general definition. Definition A.1. [[LABEL D:1006 Ord. Signed Graph]] An ordinary signed graph is a signed ordinary graph, that is, Σ = (Γ; σ) = (V; E; σ) where Γ is an ordinary graph (its edges are links and loops) and σ : E +; is any function. ! f −} Definition A.2. [[LABEL D:1006 Signed Graph]] For any graph Γ, which may have half or loose edges, we define E∗ = e : E(Γ) : e is a loop or link .A signed graph is Σ = (Γ; σ) = (V; E; σ) where Γ is any graphf and σ is any function σ : Eg∗ +; . ! f −} In either case, we call σ the (edge) signature or sign function. Not surprisingly, we refer + + to +; as the sign group. One may instead use the additive group Z2 = F2 = 0; 1 as thef sign−} group, or the group of signed units +1; 1 . We prefer the strictly multiplicativef g point of view implied by +; for reasons thatf will− becomeg clear when we discuss equations (Section I); a hint appearsf when−} we define the sign of a walk. 0 0 0 0 A subgraph Γ = (V ;E ) of Γ is naturally signed by the signature σ = σ E0 . An edge j subset S E makes the natural signed spanning subgraph (V; S; σ S). ⊆ j Definition A.3. [[LABEL D:1006 Isomorphism]] Signed graphs Σ1 and Σ2 are isomorphic if there is a graph isomorphism between Σ1 and Σ2 that preserves edge signs. Many people write +1 and 1 instead of + and . This is harmless as long as we remember the symbols are not numbers− to be added. (I will− eat these words when it comes to defining the adjacency matrix.) To some signed graph theorists (in particular, Slilaty), loose edges are positive, and half edges are negative. This is not a convention I will use. For general culture, I point out that it is well known that graph theory has been invented independently by many people. Signed graph theory was also independently invented by multiple people for multiple purposes. A curious sidelight is that, in knot theory, there is a similar-looking assignment of labels +1; 1 to edges. This is not a signed graph in our sense because the \signs" +1 and 1 f − g − 1For benefit of vegetarians: the term \meat" is intended in its early sense of `substantial food', not ‘flesh’. Section A.4 3 are interchangeable, so they do not form a group. The sign group +; is present implicitly through the action of swapping or not swapping the edge labels.f −} A.2. Examples of signed graphs. [[LABEL 2.examples]] The restriction of Σ to an edge set S is Σ S := (V; S; σ S). Oct 22: Several different signed graphs can be constructedj fromj an ordinary graph Γ. Jackie Kaminski Example A.1. [[LABEL X:1022all+]] +Γ = (Γ; +), where all edges are positive. We call it the all-positive Γ. Example A.2. [[LABEL X:1022all-]] Γ = (Γ; ), where all edges are negative. Unsurpris- ingly, we call it the all-negative Γ. − − Example A.3. [[LABEL X:1022homogeneous]] If Σ is all positive or all negative we call it homogeneous. Otherwise it is heterogeneous.2 Example A.4. [[LABEL X:1022pmG]] Γ = (+Γ) ( Γ), where we differentiate the positive and negative edges, i.e., e Γ± +e; e [ −Γ. Thus, V ( Γ) = V (Γ) and E( Γ) = +(E(Γ)) (E(Γ)). We call2 this7! the signed− 2 expansion ± of Γ. Loops± and especially half± edges can be problematic[· − here, so we generally would assume Γ is a link graph. Example A.5. [[LABEL X:1022full]] Σ is full if every vertex supports a half edge or a negative loop. In other words, referring ahead to Section A.5, every vertex supports an unbalanced edge. This is not the same as having every vertex supporting a negative edge. Note that the terms positive and balanced, or negative and unbalanced, are equivalent for circles, but not for edges. Example A.6. [[LABEL X:1022filled]] Σ◦ is Σ with a negative loop adjoined to every vertex that doesn't have a negative loop or half edge. Σ• is Σ with a negative loop or half edge adjoined to every vertex that doesn't have one. We can also define +Γ◦, Γ◦, and Γ◦. We think of doing the +, , or before adding the negative loops, so that the− final result± has one negative loop at each− vertex,± not a positive loop or two loops.