A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

of (1981a). (sg: LG, A(LG), Aut(LG)) 1981a Generalized line graphs. J. Graph Theory 5 (1981), 385–399. MR 82k:05091. Zbl. 475.05061. (sg: LG, A(LG), Aut(LG)) Dragoš M. Cvetković and Slobodan K. Simić 1978a Graphs which are switching equivalent to their line graphs. Publ. Inst. Math. (Beograd) (N.S.) 23 (37) (1978), 39–51. MR 80c:05108. Zbl. 423.05035. (sw: LG) E. Damiani, O. D’Antona, and F. Regonati 1994a Whitney numbers of some geometric lattices. J. Combin. Theory Ser. A 65 (1994), 11–25. MR 95e:06019. Zbl. 793.05037. Dowling lattices are an example. (gg: M: N) the electronic journal of combinatorics #DS8 36 O. D’Antona See E. Damiani. George B. Dantzig 1963a Linear Programming and Extensions. Princeton Univ. Press, Princeton, N.J., 1963. MR 34 #1073. Zbl. (e: 108.33103). Chapter 21: “The weighted distribution problem.” 21-2: “Linear graph structure of the basis.” (GN: M(Bases)) Prabir Das and S.B. Rao 1983a Alternating eulerian trails with prescribed degrees in two edge-colored complete graphs. Discrete Math. 43 (1983), 9–20. MR 84k:05069. Zbl. 494.05020. Given an all-negative bidirected Kn and a positive integer fi = 2gi for each vertex vi . There is a connected subgraph having in-degree and out-degree = gi at vi iff there is a g -factor of introverted and one of extroverted edges and the degrees satisfy a complicated degree condition. Generalizes Thm. 1 of Bánkfalvi and Bánkfalvi (1968a). [See Bang-Jensen and Gutin (1997a) for how to convert an edge 2-coloring to an orientation of an all-negative graph and for further developments on alternating walks.] (p: o) James A. Davis 1963a Structural balance, mechanical solidarity, and interpersonal relations. Amer. J. Sociology 68 (1963), 444–463. Reprinted with minor changes in: Joseph Berger, Morris Zelditch, Jr., and Bo Anderson, eds., Sociological Theories in Progress, Vol. One, Ch. 4, pp. 74–101. Houghton Mifflin, Boston, 1966. Also reprinted in: Samuel Leinhardt, ed., Social Networks: A Developing Paradigm, pp. 199–217. Academic Press, New York, 1977. (PsS: SG, WG: Exp) 1967a Clustering and structural balance in graphs. Human Relations 20 (1967), 181–187. Reprinted in: Samuel Leinhardt, ed., Social Networks: A Developing Paradigm, pp. 27–33. Academic Press, New York, 1977. James A. Davis and Samuel Leinhardt 1972a The structure of positive interpersonal relations in small groups. In: Joseph Berger, Morris Zelditch, Jr., and Bo Anderson, eds., Sociological Theories in Progress, Vol. Two, Ch. 10, pp. 218–251. Houghton Mifflin, Boston, 1972. Analysis of a sociological theory incorporating structural balance in relation to both randomly generated and observational data. (PsS: SG) A.C. Day, R.B. Mallion, and M.J. Rigby 1983a On the use of Riemannian surfaces in the graph-theoretical representation of Möbius systems. In: R.B. King, ed., Chemical Applications of Topology and Graph Theory (Proc. Sympos., Athens, Ga., 1983), pp. 272–284. Stud. Physical Theoret. Chem., 28. Elsevier, Amsterdam, 1983. MR 85h:05039. A clumsy but intriguing way of representing some signed (or more generally, Zn -weighted) graphs: via 2-page (or, n -page) looseleaf book embedding (all vertices are on the spine and each edge is in a single page), with an edge in page k weighted by the “sheet parity index” αk = (−1) (or, e ). (Described in the [unnecessary] terminology of an n -sheeted Riemann surface.) [A Zn -weighted) graph has such a representation iff the subgraph of edges with each weight is outerplanar.] A variation to get switching classes of signed polygons: replace αk by the “connectivity parity index” αk k where σk = number of edges in page k . the electronic journal of combinatorics #DS8 37 [The variation is valid only for polygons.] [Questions vaguely suggested by these procedures: Which signed graphs can be switched so that the edges of each sign form an outerplanar graph? Also, the same for gain graphs. And there are many similar questions: for instance, the same ones with “outerplanar” replaced by “planar.”] (SG: sw, A, T, Chem: Exp, Ref)(WG: A, T: Exp, Ref) Anne Delandtsheer 1995a Dimensional linear spaces. In: F. Buekenhout, ed., Handbook of Incidence Geometry: Buildings and Foundations, Ch. 6, pp. 193–294. North-Holland, Amsterdam,1995. MR 96k:51012. Zbl. 950.23458. “Dimensional linear space” (DLS) = simple matroid. §2.7: “Dowling lattices,” from Dowling (1973b). §6.7: “Subgeometry-closed and hereditary classes of DLS’s,” from Kahn and Kung (1982a). In §2.6, the “Enough modular hyperplanes theorem” from Kahn and Kung (1986a). (GG: M: Exp) John G. del Greco See also C.R. Coullard. 1992a Characterizing bias matroids. Discrete Math. 103 (1992), 153–159. MR 93m:05050. Zbl. 753.05021. How to decide, given a matroid M and a biased graph Ω, whether M = G(Ω). (GG: M) B. Derrida, Y. Pomeau, G. Toulouse, and J. Vannimenus 1979a Fully frustrated simple cubic lattices and the overblocking effect. J. Physique 40 (1979), 617–626. (SG: Phys, Fr) 1980a Fully frustrated simple cubic lattices and phase transitions. J. Physique 41 (1980), 213–221. MR 80m:82020. (Phys: SG) Michel Marie Deza and Monique Laurent 1997a Geometry of Cuts and Metrics. Algorithms and Combinatorics, Vol. 15. Springer, Berlin, 1997. MR 98g:52001. Zbl. 885.52001. A main object of interest is the cut polytope, which is the bipartite subgraph polytope (see Barahona, Grötschel, and Mahjoub (1985a)) of Kn , i.e. the balanced subgraph polytope (Poljak and Turźık (1987a)) of −Kn . §4.5, “An application to statistical physics”, briefly discusses the spin glass application. §26.3, “The switching operation”, discusses graph switching and its generalization to sets. §30.3, “Circulant inequalities”, mentions Poljak and Turźık (1987a, 1992a). No explicit mention of signed graphs. (p: fr: G: Exp) Persi Diaconis See K.S. Brown. V. Di Giorgio 1974a 2-modules dans un graphe: equilibre et coequilibre d’un bigraphe—application taxonomique. Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 18 (66) (1974), 81–102 (1975). MR 57 #16124. Zbl. 324.05127. (SG: B) Yvo M.I. Dirickx and M.R. Rao 1974a Networks with gains in discrete dynamic programming. Management Sci. 20 (1974), No. 11 (July, 1974), 1428–1431. MR 50 #12279. Zbl. 303.90052. (GN: M(bases)) the electronic journal of combinatorics #DS8 38 Michael Doob See also D.M. Cvetković. 1970a A geometric interpretation of the least eigenvalue of a line graph. In: Proc. Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications (1970), pp. 126–135. Univ. of North Carolina at Chapel Hill, Chapel Hill, N.C., 1970. MR 42 #2959. Zbl. 209, 554 (e: 209.55403). A readable, tutorial introduction to (1973a) (without matroids). (ec: LG, I, A(LG)) 1973a An interrelation between line graphs, eigenvalues, and matroids. J. Combin. Theory Ser. B 15 (1973), 40–50. MR 55 #12573. Zbl. 245.05125, (257.05132). Along with Simões-Pereira (1973a), introduces to the literature the evencycle matroid G(−Γ) [previously invented by Tutte, unpublished]. The multiplicity of −2 as an eigenvalue (in characteristic 0) equals the number of independent even polygons = n − rkG(−Γ). In characteristic p there is a similar theorem, but the pertinent matroid is G(Γ) if p = 2 and, when p|n , the matroid has rank 1 greater than otherwise [a fact that mystifies me]. (EC: LG, I, A(LG)) 1974a Generalizations of magic graphs. J. Combin. Theory Ser. B 17 (1974), 205–217. MR 51 #274. Zbl. 271.05128, (287.05124). (ec: I) 1974b On the construction of magic graphs. In: F. Hoffman et al., eds., Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1974), pp. 361–374. Utilitas Math. Publ. Inc., Winnipeg, Man., 1974. MR 53 #13039. Zbl. 325.05123. (ec: I) 1978a Characterizations of regular magic graphs. J. Combin. Theory Ser. B 25 (1978), 94–104. MR 58 #21840. Zbl. 384.05054. (ec: I) Michael Doob and Dragoš Cvetković 1979a On spectral characterizations and embeddings of graphs. Linear Algebra Appl. 27 (1979), 17–26. MR 81d:05050. Zbl. 417.05025. (sg: LG, A(LG)) Patrick Doreian, Roman Kapuscinski, David Krackhardt, and Janusz Szczypula 1996a A brief history of balance through time. J. Math. Sociology 21 (1996), 113–131. Reprinted in Patrick Doreian and Frans N. Stokman, eds., Evolution of Social Networks, pp. 129–147. Gordon and Breach, Australia, Amsterdam, etc., 1997. §2.3: “A method for group balance”. Describes the negation-minimal index of clusterability (generalized imbalance) from Doreian and Mrvar (1996a). (SG: B, Cl: Fr(Gen): Exp) §3.3: “Results for group balance”. Describes results from analysis of data on a small (social) group, in terms of frustration index l and a clusterability index mink>2 2Pk,.5 (slightly different from the index in Doreian and Mrvar (1996a)), finding both measures (but more so the latter) decreasing with time. (PsS: B, Cl: Fr(Gen)) Patrick Doreian and Andrej Mrvar 1996a A partitioning approach to structural balance. Social Networks 18 (1996), 149– 168. the electronic journal of combinatorics #DS8 39 They propose indices for clusterability that generalize the frustration index. Fix k ≥ 2 and α ∈ [0, 1]. For a partition π of V into k parts, they define P (π) := αn− + (1 − α)n+ , where n+ := |E+〈π〉| = number of positive edges between parts and n− := |E− :π| = number of negative edges within parts. The first proposed measure is minP (π), minimized over k partitions. [Call this Pk,α .] A second suggestion is the “negation-minimal index of generalized imbalance [i.e., of clusterability]”, the smallest number of edges whose negation (equivalently, deletion) makes Σ clusterable; it = mink 2Pk,.5 . [Note that P (π) effectively generalizes the Potts Hamiltonian as given by Welsh (1993a). Question. Does P (π) fit into an interesting generalized Potts model?] [P (π) also resembles the Potts Hamiltonian in Fischer and Hertz (1991a) (q.v. for a related research question).] They employ a local optimization algorithm to evaluate Pk,α and find an optimal partition: random descent from partition to neighboring partition, where π and π′ are neighbors if they differ by transfer of one vertex or exchange of two vertices between two parts. This was found to work well if repeated many times. [A minimizing partition into at most k parts is equivalent to a ground state of the k -spin Potts model in the form given by Welsh (1993a), but not quite of that in Fischer and Hertz (1991a).] Terminology: P (π) is called the “criterion function” [more explicitly, one might call it the ‘clusterability (adjusted by α)’ of π ]; clusterability is “k balance” or “generalized balance”. The partition’s parts are “plus-sets”. Signed digraphs are employed in the notation but direction is ignored. (SD: sg: B, Cl: Fr(Gen), Alg, PsS) 1996b Structural balance and partitioning signed graphs. In: A. Ferligoj and A. Kramberger, eds., Developments in Data Analysis, pp. 195–208. Metodološki zvezki, Vol. 12. FDV, Ljubljana, Slovenia, 1996. Similar to (1996a). Some lesser theoretical detail; some new examples. The k -clusterability index Pk,α (see (1996a)) is compared for different values of k , seeking the minimum. [But for which value(s) of α is not stated.] Interesting observation: optimal values of k were small. It is said that positive edges between parts are far more acceptable socially than negative edges within parts [thus, in the criterion function α should be rather near 1]. (SD: sg: B, Cl: Fr(Gen), Alg, PsS) W. Dörfler 1977a Double covers of graphs and hypergraphs. In: Beitrage zur Graphentheorie und deren Anwendungen (Proc. Internat. Colloq., Oberhof, D.D.R., 1977), pp. 67–79. Technische Hochschule, Ilmenau, 1977. MR 82c:05074. Zbl. 405.05055. (SG: Cov, LG)(SD, S(Hyp): Cov) 1978a Double covers of hypergraphs and their properties. Ars Combinatoria 6 (1978), 293–313. MR 82d:05085. Zbl. 423.050532. (S(Hyp): Cov, LG) Lynne L. Doty See F. Buckley. Peter Doubilet 1971a Dowling lattices and their multiplicative functions. In: Möbius Algebras (Proc. Conf., Waterloo, Ont., 1971), pp. 187–192. Univ. of Waterloo, Ont., 1971, rethe electronic journal of combinatorics #DS8 40 printed 1975. MR 50 #9605. Zbl. 385.05008. (GG: M) Peter Doubilet, Gian-Carlo Rota, and Richard Stanley 1972a On the foundations of combinatorial theory (VI): The idea of generating function. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, Calif., 1970/71), Vol. II: Probability Theory, pp. 267–318. Univ. of California Press, Berkeley, Calif., 1972. MR 53 #7796. Zbl. 267.05002. Reprinted in: Gian-Carlo Rota, Finite Operator Calculus, pp. 83–134. Academic Press, New York, 1975. MR 52 #119. Zbl. 328.05007. Reprinted again in: Joseph P.S. Kung, ed., Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, pp. 148–199. Birkhäuser, Boston, 1995. MR 99b:01027. Zbl. 841.01031. Section 5.3: Brief gain-graphic treatment of Dowling lattices. (GG: M) T.A. Dowling 1971a Codes, packings, and the critical problem. In: Atti del Convegno di Geometria Combinatoria e sue Applicazioni (Perugia, 1970), pp. 209–224. Ist. Mat., Univ. di Perugia, Perugia, Italy, 1971. MR 49 #2438. Zbl. 231.05029. Pp. 221–223: The first intimations of Dowling lattices/geometries/matroids, as in (1973a, 1973b), and their higher-weight relatives (see Bonin 1993a). (gg, Gen: M) 1973a A q -analog of the partition lattice. Ch. 11 in: J. N. Srivastava et al., eds., A Survey of Combinatorial Theory (Proc. Internat. Sympos., Ft. Collins, Colo., 1971), pp. 101–115. North-Holland, Amsterdam, 1973. MR 51 #2954. Zbl. 259.05023. Linear-algebraic progenitor of (1973b). Treats the Dowling lattice of group GF(q)× as naturally embedded in PGn−1(q). Interesting is p. 105, Remark: One might generalize some results to any ambient (simple) matroid. (gg: M, N, GG) ††1973b A class of geometric lattices based on finite groups. J. Combin. Theory Ser. B 14 (1973), 61–86. MR 46 #7066. Zbl. 247.05019. Erratum. Ibid. 15 (1973), 211. MR 47 #8369. Zbl. 264.05022. Introduces the Dowling lattices of a group, treated as lattices of grouplabelled partial partitions. Equivalent to the bias matroid of complete G -gain graph GK• n . [The gain-graphic approach was known to Dowling (1973a, p. 109) but first published in Doubilet, Rota, and Stanley (1972a).] Isomorphism, vector representation, Whitney numbers and characteristic polynomial. [The first and still fundamental paper.] (gg: M, N) Pauline van den Driessche See van den Driessche (under ‘V’). J.M. Drouffe See R. Balian. Richard A. Duke, Paul Erdös, and Vojtěch Rödl 1992a Cycle-connected graphs. Discrete Math. 108 (1992), 261–278. MR 94a:05106. Zbl. 776.05057. All graphs are simple. This is one of four related papers that prove extremal results concerning subgraphs of −Γ within which every two edges belong to a balanced polygon of length at most 2k , for all or particular k . Typical theorem: Let Fl(n,m) = the largest number m′ = m′(n,m) such that every −Γ with |V | = n and |E| ≥ m has a subgraph Σ′ with |E′| = m′ in which the electronic journal of combinatorics #DS8 41 every two edges belong to a balanced polygon of length at most l . For m = m(n) ≥ n , there is a constant c3 > 0 such that Fl(n,m) ≤ c3mn for all l . (§2, (2).) [Problem. Extend these extremal results in an interesting way to arbitrary signed simple graphs, or to simply signed graphs (no repeated edges with the same sign). (Merely allowing positive edges in addition to negative ones just makes the problem easier. Something more is required.)] (p: b(Polygons): X) Arne Dür 1986a Möbius Functions, Incidence Algebras and Power Series Representations. Lecture Notes in Math., Vol. 1202. Springer-Verlag, Berlin, 1986. MR 88m:05005. Zbl. 592.05006. Dowling lattices are an example of a categorial approach to incidence-algebra techniques in Ch. IV, §7. Computed are the characteristic polynomial and second kind of Whitney numbers. Binomial concavity, hence unimodality of the latter [cf. Stonesifer (1975a)] is proved by showing that a suitable generating polynomial has only distinct, negative roots [cf. Benoumhani (1999a)]. (gg: M: N) Paul H. Edelman and Victor Reiner 1994a Free hyperplane arrangements between An−1 and Bn . Math. Z. 215 (1994), 347– 365. MR 95b:52021. Zbl. 793.05122. Characterizes all Σ ⊇ +Kn whose bias matroid G(Σ) is supersolvable, free, or inductively free. Essentially, iff the negative links form a threshold graph. [Continued in Bailey (20xxa). Generalized in part to arbitrary gain groups in Zaslavsky (20xxh).] (sg: M, G, col) 1996a Free arrangments and rhombic tilings. Discrete Computat. Geom. 15 (1996), 307– 340. MR 97f:52019. Zbl. 853.52013. Erratum. Discrete Computat. Geom. 17 (1997), 359. MR 97k:52013. Zbl. 853.52013. Paul H. Edelman and Michael Saks 1979a Group labelings of graphs. J. Graph Theory 3 (1979), 135–140. MR 80j:05071. Zbl. 411.05059. Given Γ and abelian group A . Vertex and edge labellings λ : V → A and η : E → A are “compatible” if λ(v) = ∑ e η(e) for every vertex v , the sum taken over all edges incident with v . λ is “admissible” if it is compatible with some η . Admissible vertex labellings are characterized (differently for bipartite and nonbipartite graphs) and the number of edge labelings compatible with a given vertex labelling is computed. [Dual in a sense to Gimbel (1988a).] (WG, VS: B(D), E) Jack Edmonds See also J. Aráoz and E.L. Lawler (1976a). 1965a Paths, trees, and flowers. Canad. J. Math. 17 (1965), 449–467. MR 31 #2165. Zbl. 132, 209 (e: 132.20903). Followed up by much work, e.g., Witzgall and Zahn (1965a); see Ahuja, Magnanti, and Orlin (1993a) for some references. (p: o: i, Alg) 1965b Maximum matching and a polyhedron with 0, 1-vertices. J. Res. Nat. Bur. Standards (U.S.A.) Sect. B 69B (1965), 125–130. MR 32 #1012. Zbl. (e: 141.21802). Alludes to the polyhedron of Edmonds and Johnson (1970a). (p: o: I, G) the electronic journal of combinatorics #DS8 42 Jack Edmonds and Ellis L. Johnson ††1970a Matching: a well-solved class of integral linear programs. In: Richard Guy et al., eds., Combinatorial Structures and Their Applications (Proc. Calgary Internat. Conf., Calgary, 1969), pp. 89–92. Gordon and Breach, New York, 1970. MR 42 #2799. Zbl. 258.90032. Introduces “bidirected graphs”. A “matching problem” is an integer linear program with nonnegative and possibly bounded variables and otherwise only equality constraints, whose coefficient matrix is the incidence matrix of a bidirected graph. No proofs. [See Aráoz, Cunningham, Edmonds, and Green-Krótki (1983a) for further work.] (sg: O: I, Alg, G) Richard Ehrenborg and Margaret A. Readdy 1998a On valuations, the characteristic polynomial, and complex subspace arrangements. Advances Math. 134 (1998), 32–42. MR 98m:52018. Zbl. 906.52004. An abstract additive approach to the characteristic polynomial, applied in particular to “divisor Dowling arrangements” of hyperplanes and certain interpolating arrangements. Let Φ = G1K1 ∪ · · · ∪ GnKn , where V (Ki) = {v1, . . . , vi} and G1 ≥ · · · ≥ Gn is a chain of subgroups of a gain group G = G1 . When G is finite cyclic, the complex hyperplane representation of Φ• is a “divisor Dowling arrangement”. [Its polynomial equals the chromatic polynomial of Φ• , which is easily computed via gain-graph coloring without the restriction to cyclic gain group. The same appears to be true for the other arrangements treated herein.] (gg: M: G, N) 1999a On flag vectors, the Dowling lattice, and braid arrangements. Discrete Computat. Geom. 21 (1999), 389–403. The Dowling lattice is that of a finite cyclic group Zk . Thm. 4.9 is a recursive formula for its flag h-vector (in the form of the ab -index). Thm. 5.2 is a similar formula for the c, 2d-index of the face lattices of the real root system arrangements An and Bn , whose intersection lattices are the Dowling lattices of Z1 and Z2 . §6 presents a combinatorial description of the face lattice of Bn [which it is interesting to compare with that in Zaslavsky (1991b)]. (gg: M: G, N) A. Ehrenfeucht, T. Harju, and G. Rozenberg 1996a Group based graph transformations and hierarchical representations of graphs. In: J. Cuny, H. Ehrig, G. Engels and G. Rozenberg, eds., Graph Grammars and Their Application to Computer Science (5th Internat. Workshop, Williamsburg, Va., 1994), pp. 502–520. Lecture Notes in Computer Science, Vol. 1073. SpringerVerlag, Berlin, 1996. MR 97h:68097. The “heierarchical structure” of a switching class of skew gain graphs based on Kn . (gg: K: Sw) 1997a 2-Structures—A framework for decomposition and transformation of graphs. In: Grzegorz Rozenberg, ed., Handbook of Graph Grammars and Computing by Graph Transformation. Vol. 1: Foundations, Ch. 6, pp. 401–478. World Scientific, Singapore, 1997. MR 99b:68006 (book). Zbl. 908.68095 (book). A tutorial (with some new proofs). §6.7: “Dynamic labeled 2-structures”. §6.8: “Dynamic `2-structures with variable domains”. §6.9: “Quotients and plane trees”. §6.10: “Invariants”. (gg: sw: Exp, Ref) 1997b Invariants of inversive 2-structures on groups of labels. Math. Structures Computer Sci. 7 (1997), 303–327. MR 98g:20089. Zbl. 882.05119. the electronic journal of combinatorics #DS8 43 Given a gain graph (Kn, φ,G), a word w in the oriented edges of Kn has a gain φ(w); call this ψw(φ). A “free invariant” is a ψw that is an invariant of switching classes. Thm.: There is a number d = d(Kn,G) such that the group of free invariants is generated by ψw with w = z 1 · · · z ku1 · · ·ul where wi are triangular cycles (directed!) and ui are commutators. [The whole paper applies mutatis mutandis to arbitrary graphs, the triangular cycles being replaced by any set of cycles containing a fundamental system.] Dictionary: “Inversive 2-structure” = gain graph based on Kn . (gg: K: Sw, N) Andrzej Ehrenfeucht and Grzegorz Rozenberg 1993a An introduction to dynamic labeled 2-structures. In: Andrzej M. Borzyszkowski and Stefan Soko lowski, eds., Mathematical Foundations of Computer Science 1993 (Proc., 18th Internat. Sympos., MFCS ’93, Gdańsk, 1993), pp. 156–173. Lecture Notes in Computer Sci., Vol. 711. Springer-Verlag, Berlin, 1993. MR 95j:68126. Extended summary of (1994a). (GG(Gen): K: Sw, Str) 1994a Dynamic labeled 2-structures. Math. Structures Comput. Sci. 4 (1994), 433–455. MR 96j:68144. Zbl. 829.68099. They prove that a complicated definition of “reversible dynamic labeled 2structure” G amounts to a complete graph with a set, closed under switching, of twisted gains in a gain group ∆. The twist is a gain-group automorphism α such that λ(e;x, y) = [αλ(e; y, x)]−1 , λ being the gain function. Dictionary: their “domain” D = vertex set, “labeling function” λ (or equivalently, g ) = gain function, “alphabet” = gain group, “involution” δ = α◦ inversion, “δ -selector” Ŝ = switching function, “transformation induced by Ŝ ” = switching by Ŝ ; a “single axiom” d.l. 2-structure consists of a single switching class. Further, they investigate “clans” of G . Given g (i.e., λ), deleting identitygain edges leaves isolated vertices (“horizons”) and forms connected components, any union of which is a “clan” of g . A clan of G is any clan of any g ∈ G . (GG(Gen): K: Sw, Str) 1994b Dynamic labeled 2-structures with variable domains. In: J. Karhumäki, H. Maurer, and G. Rozenberg, eds., it Results and Trends in Theoretical Computer Science (Proc., Colloq. in Honor of Arto Alomaa, Graz, 1994), pp. 97–123. Lecture Notes in Computer Science, Vol. 812. Springer-Verlag, Berlin, 1994. MR 95m:68128. Combinations and decompositions of complete graphs with twisted gains. (GG(Gen): K: Str, Sw) Kurt Eisemann 1964a The generalized stepping stone method for the machine loading model. Management Sci. 11 (1964/65), No. 1 (Sept., 1964), 154–176. Zbl. 136, 139 (e: 136.13901). (GN: I, M(bases)) Joyce Elam, Fred Glover, and Darwin Klingman 1979a A strongly convergent primal simplex algorithm for generalized networks. Math. Oper. Res. 4 (1979), 39–59. MR 81g:90049. Zbl. 422.90081. (GN: M(bases), I) David P. Ellerman 1984a Arbitrage theory: A mathematical introduction. SIAM Rev. 26 (1984), 241–261. MR 85g:90024. Zbl. 534.90014. (GG: B, I, Flows: Appl, Ref) the electronic journal of combinatorics #DS8 44 M.N. Ellingham 1991a Vertex-switching, isomorphism, and pseudosimilarity. J. Graph Theory 15 (1991), 563–572. MR 92g:05136. Zbl. 802.05057. Main theorem (§2) characterizes, given two signings of Kn (where n may be infinite) and a vertex set S , when switching S makes the signings isomorphic. [Problem 1. Generalize to other underlying graphs. Problem 2. Prove an analog for bidirected Kn ’s.] A corollary (§3) characterizes when vertices u, v of Σ = (Kn, σ) satisfy Σ{u} ∼= Σ{v} and discusses when in addition no automorphism of Σ moves u to v . All is done in terms of Seidel (graph) switching (here called “vertex-switching”) of unsigned simple graphs. (k: sw, TG) 1996a Vertex-switching reconstruction and folded cubes. J. Combin. Theory Ser. B 66 (1966), 361–364. MR 96i:05120. Zbl. 856.05071. Deepens the folded-cube theory of Ellingham and Royle (1992a). Nicely generalizing Stanley (1985a), the number of subgraphs of a signed Kn that are isomorphic to a fixed signed Km is reconstructible from the s-vertex switching deck if the Krawtchouk polynomial K s (x) has no even zeros between 0 and m . (Closely related to Krasikov and Roditty (1992a), Theorems 5 and 6.) Remark 4: balance equations (Krasikov and Roditty (1987a)) and Krawtchouk polynomials both reflect properties of folded cubes. All is done in terms of Seidel switching of unsigned simple graphs. [It seems clear that the folded cube appears because it corresponds to the effect of switchings on signatures of Kn (or any connected graph), since switching by X and X have the same effect. For the bidirected case (Problem 2 under Stanley (1985a)), the unfolded cube should play a similar role. Question. When treating a general underlying graph Γ, will a polynomial influenced by Aut Γ replace the Krawtchouk polynomial?] (k: sw, TG) M.N. Ellingham and Gordon F. Royle 1992a Vertex-switching reconstruction of subgraph numbers and triangle-free graphs. J. Combin. Theory Ser. B 54 (1992), 167–177. MR 93d:05112. Zbl. 695.05053 (748.05071). Reconstruction of induced subgraph numbers of a signed Kn from the s vertex switching deck, dependent on linear transformation and thence Krawtchouk polynomials as in Stanley (1985a). The role of those polynomials is further developed. Done in terms of Seidel switching of unsigned simple graphs, with the advantage of reconstructing arbitrary subgraph numbers as well. A gap is noted in Krasikov and Roditty (1987a), proof of Lemma 2.5. [Methods and results are closely related to Krasikov (1988a) and Krasikov and Roditty (1987a, 1992a).] (k: sw, TG) Gernot M. Engel and Hans Schneider 1973a Cyclic and diagonal products on a matrix. Linear Algebra Appl. 7 (1973), 301–335. MR 48 #2160. Zbl. 289.15006. (gg: Sw) 1975a Diagonal similarity and equivalence for matrices over groups with 0. Czechoslovak Math. J. 25 (100) (1975), 389–403. MR 53 #477. Zbl. 329.15007. (gg: Sw) 1980a Matrices diagonally similar to a symmetric matrix. Linear Algebra Appl. 29 (1980), 131–138. MR 81k:15017. Zbl. 432.15014. (gg: Sw) the electronic journal of combinatorics #DS8 45 R.C. Entringer 1985a A short proof of Rubin’s block theorem. In: B.R. Alspach and C.D. Godsil, eds., Cycles in Graphs, pp. 367–368. Ann. Discrete Math., Vol. 27. North-Holland Math. Stud., Vol. 115. North-Holland, Amsterdam, 1985. MR 87f:05144. Zbl. 576.05037. See Erdős, Rubin, and Taylor (1980a). (p: b) H. Era See J. Akiyama. Pál Erdős [sometimes, Paul Erdös] See also B. Bollobás and R.A. Duke. 1996a On some of my favourite theorems. In: D. Miklós, V.T. Sós and T. Szőnyi, eds., Combinatorics, Paul Erdős is Eighty (Papers from the Internat. Conf. on Combinatorics, Keszthely, 1993), Vol. 2, pp. 97–132. Bolyai Soc. Math. Studies, 2. János Bolyai Mathematical Society, Budapest, 1996. MR 97g:00002. Zbl. 837.00020 (book). P. 119 mentions the theorem of Duke, Erdős, and Rödl (1991a) on even polygons. Pp. 120–121 mention (amongst similar problems) a theorem of Erdős and Hajnal (source not stated): Every all-negative signed graph with chromatic number א1 contains every finite bipartite graph [i.e., every finite, balanced, all-negative signed graph]. [Problem. Find generalizations to signed graphs. For instance: Conjecture. Every signed graph with chromatic number א1 , that does not become antibalanced upon deletion of any finite vertex set, contains every finite, balanced signed graph up to switching equivalence.] [The MR review: “this is one of the best collections of problems that Erdos has published.”] (p: b: Exp, Ref) P. Erdös, R.J. Faudree, A. Gyárfás, and R.H. Schelp 1991a Odd cycles in graphs of given minimum degree. In: Y. Alavi, G. Chartrand, O.R. Oellermann, and A.J. Schenk, eds., Graph Theory, Combinatorics, and Applications (Proc. Sixth Quadren. Internat. Conf. Theory Appl. Graphs, Kalamazoo, Mich., 1988), Vol. 1, pp. 407–418. Wiley, New York, 1991. MR 93d:05085. Zbl. 840.05050. A large, nonbipartite, 2-connected graph with large minimum degree contains a polygon of given odd length or is one of a single type of exceptional graph. [Question. Can this be generalized to negative polygons in unbalanced signed graphs?] (p, sg: Polygons, X) P. Erdős, E. Győri, and M. Simonovits 1992a How many edges should be deleted to make a triangle-free graph bipartite? In: G. Halász, L. Lovász, D. Miklós, and T. Szönyi, eds., Sets, Graphs and Numbers (Proc., Budapest, 1991), pp. 239–263. Colloq. Math. Soc. János Bolyai, Vol. 60. János Bolyai Math. Soc., Budapest, and North-Holland, Amsterdam, 1992. MR 94b:05104. Zbl. 785.05052. Assume |Σ| simple of order n and + a fixed graph ∆. Results on frustration index l of antibalanced Σ if ∆ is 3-chromatic, esp. C3 . Thm.: If |E| > n/5− o(n), then l(Σ) < n/25− o(n). Conjecture (Erdős): For ∆ = C3 the hypothesis on |E| is unnecessary. [Question 1(a). Is the answer different when Σ need not be antibalanced? Question 2(a). Exclude a fixed signed the electronic journal of combinatorics #DS8 46 graph whose signed chromatic number = 1. Question 3(a). In particular, exclude −K3 . Question 4(a). Exclude −Kl . Question 5(a). Exclude an unbalanced Cl . Questions 1–5(b). Even if l(Σ) cannot be estimated, is there always an extremal graph that is antibalanced—as when no graph is excluded, by Petersdorf (1966a)?] (p: X) Paul Erdös, Arthur L. Rubin, and Herbert Taylor 1980a Choosability in graphs. In: Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Arcata, Calif., 1979), pp. 125–157. Congressus Numer., XXVI. Utilitas Math. Publ. Inc., Winnipeg, Man., 1980. MR 82f:05038. Zbl. 469.05032. Rubin’s block theorem (Thm. R, p. 136): a block graph, not complete or an odd polygon, contains an induced even polygon with at most one chord. [See also Entringer (1985a).] [Question. Does this generalize to signed graphs, Rubin’s block theorem being the antibalanced case? Rubin’s 2-choosability theorem, p. 132, is also tantalizingly reminiscent of antibalanced graphs, but in reverse.] (p: Str, b) Cloyd L. Ezell 1979a Observations on the construction of covers using permutation voltage assignments. Discrete Math. 28 (1979), 7–20. MR 81a:05040. Zbl. 413.05005. (GG: T, Cov, sw) Arthur M. Farley and Andrzey Proskurowski 1981a Computing the line index of balance of signed outerplanar graphs. Proc. Twelfth Southeastern Conf. on Combinatorics, Graph Theory and Computing (Baton Rouge, 1981), Vol. I. Congressus Numer. 32 (1981), 323–332. MR 83m:68119. Zbl. 489.68065. Calculating frustration index is NP-complete, since it is more general than max-cut. However, for signed outerplanar graphs with bounded size of bounded faces, it is solvable in linear time. [It is quickly solvable for signed planar graphs. See Katai and Iwai (1978a), Barahona (1981a, 1982a), and more.] (SG: Fr) M. Farzan 1978a Automorphisms of double covers of a graph. In: Problemes Combinatoires et Theorie des Graphes (Colloq. Internat., Orsay, 1976), pp. 137–138. Colloques Internat. du CNRS, 260. Editions du C.N.R.S., Paris, 1978. MR 81a:05063. Zbl. 413.05064. A “double cover of a graph” means the double cover of a signing of a simple graph. (sg: Cov, Aut) R.J. Faudree See P. Erdős. Katherine Faust See S. Wasserman. N.T. Feather 1971a Organization and discrepancy in cognitive structures. Psychological Rev. 78 (1971), 355–379. A suggestion for defining balance in weighted digraphs: pp. 367–369. (PsS: B: Exp)(WD: B) the electronic journal of combinatorics #DS8 47 Lori Fern, Gary Gordon, Jason Leasure, and Sharon Pronchik 20xxa Matroid automorphisms and symmetry groups. Submitted. Consider a subgroup W of the hyperoctahedral group On that is generated by reflections. Let M(W ) be the vector matroid of the vectors corresponding to reflections in W . The possible direct factors of any automorphism group of M(W ) are Sk , Ok , and O k . The proof is stricly combinatorial, via signed graphs. (SG: Aut, G) Miroslav Fiedler 1957a Uber qualitative Winkeleigenschaften der Simplexe. Czechoslovak Math. J. 7 (82) (1957), 463–478. MR 20 #1252. Zbl. 93, 336 (e: 093.33602). (SG: G) 1957b Einige Satze aus der metrischen Geometrie der Simplexe in euklidischen Raumen. Schr. Forschungsinst. Math. 1 (1957), 157. MR 19, 303. Zbl. 89, 167 (e: 089.16706). (SG: G) 1961a Uber die qualitative Lage des Mittelpunktes der ungeschriebenen Hyperkugel im n -Simplex. Comment. Math. Univ. Carolin. 2, No. 1 (1961), 1–51. Zbl. 101, 132 (e: 101.13205). (SG: G) 1964a Some applications of the theory of graphs in matrix theory and geometry. In: Theory of Graphs and Its Applications (Proc. Sympos., Smolenice, 1963), pp. 37– 41. Publ. House Czechoslovak Acad. Sci., Prague, 1964. MR 30 #5294. Zbl. (e: 163.45605). (SG: G) 1967a Graphs and linear algebra. In: Theory of Graphs: International Symposium (Rome, 1966), pp. 131–134. Gordon and Breach, New York; Dunod, Paris, 1967. MR 36 #6313. Zbl. 263.05124. (SG: G) 1969a Signed distance graphs. J. Combin. Theory 7 (1969), 136–149. MR 39 #4034. Zbl. 181, 260 (e: 181.26001). (SG: G) 1970a Poznámka o distancnich grafech [A remark on distance graphs] (in Czech). In: Matematika (geometrie a teorie grafu) [Mathematics (Geometry and Graph Theory)], pp. 85–88. Univ. Karlova, Prague, 1970. MR 43 #3143. Zbl. 215.50203. (SG: G) 1975a Eigenvectors of acyclic matrices. Czechoslovak Math. J. 25 (100) (1975), 607–618. MR 52 #8151. Zbl. 325.15014. (sg: Trees: A) 1985a Signed bigraphs of monotone matrices. In: Horst Sachs, ed., Graphs, Hypergraphs and Applications (Proc. Internat. Conf., Eyba, 1984), pp. 36–40. Teubner-Texte zur Math., B. 73. B.G. Teubner, Leipzig, 1985. MR 87m:05121. Zbl. 626.05023. (SG: A: Exp) Miroslav Fiedler and Vlastimil Ptak 1967a Diagonally dominant matrices. Czechoslovak Math. J. 17 (92) (1967), 420–433. MR 35 #6704. Zbl. (e: 178.03402). (GG: Sw, b) 1969a Cyclic products and an inequality for determinants. Czechoslovak Math. J. 19 (94) (1969), 428–451. MR 40 #1409. Zbl. 281.15014. (gg: Sw) Joseph Fiksel 1980a Dynamic evolution in societal networks. J. Math. Sociology 7 (1980), 27–46. MR 81g:92023(q.v.). Zbl. 434.92022. (SG: Cl, VS) the electronic journal of combinatorics #DS8 48 Steven D. Fischer 1993a Signed Poset Homology and q -Analog Möbius Functions. Ph.D. thesis, Univ. of Michigan, 1993. §1.2: “Signed posets”. Definition of signed poset: a positively closed subset of the root system Bn whose intersection with its negative is empty. (Following Reiner (1990).) Equivalent to a partial ordering of ±[n] in which negation is a self-duality and each dual pair of elements is comparable. [This is really a special type of signed poset. The latter restriction does not hold in general.] Relevant contents: Ch. 2: “Cohen-Macaulay signed posets”, §2.2: “ELlabelings of posets and signed posets”, and shellability. Ch. 3: “Euler characteristics”, and a fixed-point theorem. §5.1: “The homology of the signed posets SΠ ” (a particular example). App. A: “Open problems”, several concerning signed posets. [Partially summarized by Hanlon (1996a).] (S: sg, o, G, N) K.H. Fischer and J.A. Hertz 1991a Spin Glasses. Cambridge Studies in Magnetism: 1. Cambridge Univ. Press, Cambridge, Eng., 1991. MR 93m:82019. §2.5, “Frustration”, discusses the spin glass Ising model (essentially, signed graphs) in square and cubical lattices, including the “Mattis model” (a switching of all positive signs), as well as a vector analog, the “XY” model (planar spins) and (p. 46) even a general gain-graph model with switchinginvariant Hamiltonian. From the point of view of physics (mainly theoretical physics). (Phys: SG: Fr, Sw: Exp, Ref) §3.7: “The Potts glass”. The Hamiltonian (without edge weights) is H = − 1 2 ∑ σ(eij)(kδ(si, sj) − 1). [It is not clear that the authors intend to permit negative edges. If they are allowed, H is rather like Doreian and Mrvar’s (1996a) P (π). Question. Is there a worthwhile generalized signed and weighted Potts model with Hamiltonian that specializes both to this form of H and to P ?] [Also cf. Welsh (1993a) on the Ashkin–Teller–Potts model.] (Phys: sg, cl: Exp) P.C. Fishburn and N.J.A. Sloane 1989a The solution to Berlekamp’s switching game. Discrete Math. 74 (1989), 263–290. MR 90e:90151. Zbl. 664.94024. The maximum frustration index of a signed Kt,t , which equals the covering radius of the Gale–Berlekamp code, is evaluated for t ≤ 10, thereby extending results of Brown and Spencer (1971a). See Table 1. (sg: Fr) Claude Flament 1958a L’etude mathematique des structures psycho-sociales. L’Annee Psychologique 58 (1958), 119–131. Signed graphs are treated on pp. 126–129. (SG: B, PsS: Exp) 1963a Applications of Graph Theory to Group Structure. Prentice-Hall, Englewood Cliffs, N.J., 1963. MR 28 #1014. Zbl. 141, 363 (e: 141.36301). English edition of (1965a). Ch. 3: “Balancing processes.” (SG: K: B, Alg) 1965a Theorie des graphes et structures sociales. Math. et sci. de l’homme, Vol. 2. Mouton and Gauthier-Villars, Paris, 1965. MR 36 #5018. Zbl. 169, 266 (e: 169.26603). Ch. III: “Processus d’equilibration.” (SG: K: B, Alg) the electronic journal of combinatorics #DS8 49 1970a Equilibre d’un graphe, quelques resultats algebriques. Math. Sci. Humaines, No. 30 (1970), 5–10. MR 43 #4704. Zbl. 222.05124. 1979a Independent generalizations of balance. In: Paul W. Holland and Samuel Leinhardt, eds., Perspectives on Social Network Research (Proc. Sympos., Dartmouth Coll., Hanover, N.H., 1975), Chapter 10, pp. 187–200. Academic Press, New York, 1979. (SG: B, PsS) C.M. Fortuin and P.W. Kasteleyn 1972a On the random cluster model.