A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas Compiled by Thomas Zaslavsky Manuscript prepared with Marge Pratt Department of Mathematical Sciences Binghamton University Binghamton, New York, U.S.A. 13902-6000 E-mail: [email protected] Submitted: March 19, 1998; Accepted: July 20, 1998. Sixth Edition, Revision a 20 July 1998 Mathematical Subject Classications (1991): Primary 05-02, 05C99; Secondary 05B20, 05B25, 05B35, 05C10, 05C15, 05C20, 05C25, 05C30, 05C35, 05C38, 05C45, 05C50, 05C60, 05C65, 05C70, 05C75, 05C80, 05C85, 05C90, 05E25, 05E30, 06A07, 06A09, 05C10, 15A06, 15A15, 15A39, 15A99, 20B25, 20F55, 34C11, 51D20, 51E20, 52B12, 52B30, 52B40, 52C07, 68Q15, 68Q25, 68Q35, 68R10, 82B20, 82D30, 90A08, 90B10, 90C08, 90C27, 90C35, 90C60, 92D40, 92E10, 92H30, 92J10, 92K10, 94B25. Colleagues: HELP! If you have any suggestions whatever for items to include in this bibliography, or for other changes, please let me hear from you. Thank you. Typeset by AMS-TEX Index A1 H48 O83 V 110 B8 I59 P84 W 111 C20 J60 Q90 X 118 D28 K63 R90 Y 118 E32 L71 S94 Z 115 F36 M76 T104 G39 N82 U109 Preface A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. I regard as fundamental the notion of balance of a circuit (sign product equals +, the sign group identity) and the vertex-edge incidence matrix (whose column for a negative edge has two +1’s or two 1’s, for a positive edge one +1 and one 1, the rest being zero). Hence I have included work on gain graphs (where the edge labels are taken from any group) and “consistency” in vertex-signed or marked graphs (where the vertices are signed), and generalizable work on two-graphs (the set of unbalanced triangles of a signed complete graph) and on even and odd polygons and paths in graphs and digraphs. Nevertheless, it was not always easy to decide what belongs. I have employed the following principles: Only works with mathematical content are included, except for a few (not very care- fully) selected representatives of applications papers and surveys (e.g., Taylor (1970a) and Wasserman and Faust (1993a) for psychosociology and Trinajstic (1983a) for chemistry). I do try to include: Any (mathematical) work in which signed graphs are mentioned by name or signs are put on the edges of graphs, regardless of whether it makes essential use of signs. (Exceptions: In order to maintain “balance” in the bibliography, and due to lack of time, I have included only a limited selection of items concerning signed digraphs, two-graphs, and applications. For the rst, esp. for qualitative matrix theory, see e.g. Maybee and Quirk (1969a) and Brualdi and Shader (1995a). For the second, see any of the review articles by Seidel. I have hopelessly slighted statistical physics, as a keyword search of Mathematical Reviews for “frustrat*” will plainly show.) Any work in which the notion of balance of a polygon plays a role. Example: gain graphs. (Exception: purely topological papers concerning ordinary graph embedding.) Any work in which ideas of signed graph theory are anticipated, or generalized, or transferred to other domains. Examples: marked graphs; signed posets and matroids. Any mathematical structure that is an example, however disguised, of a signed or gain graph or generalization, and is treated in a way that seems in the spirit of signed graph theory. Examples: even-cycle and bicircular matroids; bidirected graphs; some of the literature on two-graphs and double covering graphs. And some works that have suggested ideas of value for signed graph theory or that have promise of doing so in the future. Plainly, judgment is required to apply these criteria. I have employed mine freely, taking account of suggestions from my colleagues. Still I know that the bibliography is far from complete, due to the quantity and even more the enormous range and dispersion of work in the relevant areas. I will continue to add both new and old works to future editions and I heartily welcome further suggestions. There are certainly many errors, some of them egregious. For these I hand over re- sponsibility to Sloth, Pride, Ambition, Envy, and Confusion. Corrections, however, will be gratefully accepted by me. Bibliographical Data. Authors’ names are given usually in only one form, even should the name appear in dierent (but recognizably similar) forms on dierent publica- tions. Journal abbreviations follow the style of Mathematical Reviews (MR) with minor ‘improvements’. Reviews and abstracts are cited from MR and its electronic form Math- SciNet, from Zentralblatt fur Mathematik (Zbl.) and its electronic form MATH Database (For early volumes, “Zbl. VVV, PPP” denotes printed volume and page; the electronic item number is “(e VVV.PPPNN)”.), and occasionally from Chemical Abstracts (CA). A review marked (q.v.) has signicance, possibly an insight, a criticism, or a viewpoint orthogonal to mine. Some—not all—of the most fundamental works are marked with a †. This is a personal selection. Annotations. I try to describe the relevant content in a consistent terminology and notation, in the language of signed graphs despite occasional clumsiness (hoping that this will suggest generalizations), and sometimes with my [bracketed] editorial comments. I sometimes try also to explain idiosyncratic terminology, in order to make it easier to read the original item. Several of the annotations incorporate open problems (of widely varying degrees of importance and diculty). I use these standard symbols: is a graph (undirected), possibly allowing loops and multiple edges. It is normally nite unless otherwise indicated. is a signed graph. Its vertex and edge sets are V and E ; its order is n = |V |. E+ , E are the sets of positive and negative edges and + , are the corresponding spanning subgraphs (unsigned). [] is the switching class of . A( ) is the adjacency matrix. is a gain graph. is a biased graph. l( ) is the frustration index (= line index of imbalance). G( ) is the bias matroid of a signed, gain, or biased graph. Acknowledgement. I cannot name all the people who have contributed advice and criticism, but many of the annotations have beneted from suggestions by the authors or others and a number of items have been brought to my notice by helpful correspondents. I am very grateful to you all. Thanks also to the people who maintain the invaluable MR and Zbl. indices. However, I insist on my total responsibility for the nal form of all entries, including such things as my restatement of results in signed or gain graphic language and, of course, all the praise and criticism (but not errors; see above) that they contain. Subject Classication Codes Acodeinlower case means the topic appears implicitly but not explicitly. A sux w on SG, SD, VS denotes signs used as weights, i.e., treated as the numbers +1 and 1, added, and (usually) the sum compared to 0. A Adjacency matrix, eigenvalues. Alg Algorithms. Appl Applications other than (Chem), (Phys), (PsS). Aut Automorphisms, symmetries, group actions. B Balance (mathematical), cobalance. Bic Bicircular matroids. Col Vertex coloring. ECol Edge coloring. Chem Applications to chemistry. Cl Clustering. Cov Covering graphs, double coverings. D Duality (matroids or matrices). E Enumeration of types of signed graphs, etc. EC Even-cycle matroids. Exr Interesting exercises (in an expository work). Exp Expository. Fr Frustration index (line index of imbalance) and other measures of imbalance. G Connections with geometry, including toric varieties, complex complement, etc. Gen Generalization. GD Digraphs with gains (voltages). GG Gain graphs, voltage graphs, biased graphs; includes Dowling lattices. GN Generalized or gain networks. (Multiplicative real gains.) I Incidence matrix, Kirchho matrix. K Signed complete graphs. Knot Connections with knot theory (sparse coverage if signs are purely notational). LG Line graphs. M Matroids, chain-groups, ows. N Numerical and algebraic invariants of signed graphs, etc. O Orientations, bidirected graphs. OG Ordered gains. P All-negative or antibalanced signed graphs; parity-biased graphs. p Includes problems on even or odd length of paths or polygons (partial coverage). Phys Applications in physics. PsS Structural balance (psychological, sociological, and anthropological applications). QM Qualitative (sign) matrices: sign stability, sign solvability, etc. Rand Random signs or gains, signed or gain graphs. Ref Many references. S Signed objects other than graphs and hypergraphs: mathematical properties. SD Signed digraphs: mathematical properties. SG Signed graphs: mathematical properties. SH Signed hypergraphs: mathematical properties. Sol Sign solvability, sign nonsingularity (partial coverage). Sta Sign stability (partial coverage). Str Structure theory. Sw Switching of signs or gains. T Topology of the graph, surface embeddings. TG Two-graphs, graph (Seidel) switching (partial coverage). VS Vertex-signed graphs (“marked graphs”); signed vertices and edges. WD Weighted digraphs. WG Weighted graphs. 1 A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas Robert P. Abelson 1967a Mathematical models in social psychology. In: Leonard Berkowitz, ed., Advances in Experimental Social Psychology, Vol. 3, pp. 1–54. Academic Press, New York, 1967. Sect. II: “Mathematical models of social structure.” Part B: “The balance principle.” Reviews basic notions of balance and clusterability in signed (di)graphs and measures of degree of balance or clustering. Notes that signed T Kn is balanced i I +A = vv , v = 1-vector. Proposes: degree of balance = 1/n,where1 = largest eigenvalue of I + A() and n = order of the (di)graph. [Cf. Phillips (1967a).] Part C, 3: “Clusterability revisited.” (SG, SD: B, Cl, Fr, A) Robert P. Abelson and Milton J. Rosenberg 1958a Symbolic psycho-logic: a model of attitudinal cognition. Behavioral Sci. 3 (1958), 1–13. Analyzes switching via Hadamard product with “passive T -matrices”. Pro- poses frustration index (“complexity”, p. 12) l() as measure of imbalance. (Cf. Harary (1959b).) Thm. 14: max l(), over all of order n, equals b(n 1)2/4c.