Signed graphs

A signed graph is a graph with a sign attached to each arc. This article introduces the matroids of signed graphs, which generalize both the polygon matroids and the even-circle (or unoriented cycle) matroids of ordinary graphs. The concepts of balance, switching, restriction and contraction, double covering graphs, and linear representation of signed graphs are treated in terms of the matroid, and a matrix-tree theorem for signed graphs is proved. The examples treated include the all-positive and all-negative graphs (whose matroids are the polygon and even-circle matroids), sign-symmetric graphs (related to the classical root systems), and signed complete graphs (equivalent to two-graphs).Replacing the sign group by an arbitrary group leads to voltage graphs. Most of our results on signed graphs extend to all voltage graphs.     

Graph theoretic approach to parallel gene assembly

We study parallel complexity of signed graphs motivated by the highly complex genetic recombination processes in ciliates. The molecular gene assembly operations have been modeled by operations of signed graphs, i.e., graphs where the vertices have a sign + or −. In the optimization problem for signed graphs one wishes to find the parallel complexity by which the graphs can be reduced to the empty graph. We relate parallel complexity to matchings in graphs for some natural graph classes, especially bipartite graphs. It is shown, for instance, that a bipartite graph G has parallel complexity one if and only if G has a unique perfect matching. We also formulate some open problems of this research topic.     

Nowhere‐Zero Flows on Signed Complete and Complete Bipartite Graphs

Bouchet's conjecture asserts that each signed graph which admits a nowhere‐zero flow has a nowhere‐zero 6‐flow. We verify this conjecture for two basic classes of signed graphs—signed complete and signed complete bipartite graphs by proving that each such flow‐admissible graph admits a nowhere‐zero 4‐flow and we characterise those which have a nowhere‐zero 2‐flow and a nowhere‐zero 3‐flow.     

On the Laplacian Eigenvalues of Signed Graphs

A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the largest Laplacian eigenvalue of a signed graph is investigated, which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.     

Bounds for the Least Laplacian Eigenvalue of a Signed Graph

A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.     

A characterization of projective-planar signed graphs

A signed graph has a plus or minus sign on each edge. A simple cycle is positive or negative depending on whether it contains an even or odd number of negative edges, respectively. We consider embeddings of a signed graph in the projective plane for which a simple cycle is essential if and only if it is negative. We characterize those signed graphs that have such a projective-planar embedding. Our characterization is in terms of a related signed graph formed by considering the theta subgraphs in the given graph.     

Unavoidable Configurations in Complete Topological Graphs

A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least clog^1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most clog^1/6 n vertices.     

图的好符号星控制数

引入了图的好符号星控制的概念,求出了欧拉图、完全二部图、完全图和轮图的好符号星控制数,并改进了图的符号星控制数的两个上界.     

Even signings,signed switching classes,and (−1, 1)-matrices

In this paper, algebraic and combinatorial techniques are used to establish results concerning even signings of graphs, switching classes of signed graphs, and (?1, 1)-matrices. These results primarily deal with enumeration of isomorphism types, and determining whether there are fixed elements under the action of automorphisms. A formula is given for the number of isomorphism types of even signings of any fixed simple graph. This is shown to be equal to the number of isomorphism types of switching classes of signings of the graph. A necessary and sufficient criterion is found for all switching classes fixed by a given graph automorphism to contain signings fixed by that automorphism. It is determined whether this criterion is met for all automorphisms of various graphs, including complete graphs, which yields a known result of Mallows and Sloane. As an application, a formula is developed for the number of H-equivalence classes of (?1, 1)-matrices of fixed size. Independently, using Molien's theorem and following a suggestion of Cameron's, generating series for these numbers are given. As a final application, a necessary and sufficient condition that a square (?1, 1)-matrix be switching equivalent to a symmetric matrix is given.     

Isomorphic factorizations of complete equipartite graphs

A complete equipartite graph is a complete multipartite graph such that the color classes are equinumerous. An isomorphic factorization of a graph is a partition of the line set into disjoint isomorphic parts. It is shown that an isomorphic factorization of the complete equipartite graph into t isomorphic subgraphs exists whenever the total number of lines is divisible by t.     

Graph algebras and graph varieties

Graph algebras establish a connection between graphs (i.e. binary relations) and universal algebras. A structure theorem of Birkhoff-type is given which characterizes graph varieties, i.e. classes of graphs which can be defined by identities for their corresponding graph algebras: A class of finite directed graphs without multiple edges is a graph variety iff it is closed with respect to finite restricted pointed subproducts and isomorphic copies. Several applications are given, e.g., every loopless finite directed graph is an induced subgraph of a direct power of a graph with three vertices. Graphs with bounded chromatic number or density form graph varieties characterizable by identities of special kind.Presented by R. W. Quackenbush.     

Irreducible pseudo 2-factor isomorphic cubic bipartite graphs

A bipartite graph is pseudo 2-factor isomorphic if the number of circuits in each 2-factor of the graph is always even or always odd. We proved (Abreu et?al., J Comb Theory B 98:432–442, 2008) that the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graph of girth 4 is K _3,3, and conjectured (Abreu et?al., 2008, Conjecture 3.6) that the only essentially 4-edge-connected cubic bipartite graphs are K _3,3, the Heawood graph and the Pappus graph. There exists a characterization of symmetric configurations n ₃ due to Martinetti (1886) in which all symmetric configurations n ₃ can be obtained from an infinite set of so called irreducible configurations (Martinetti, Annali di Matematica Pura ed Applicata II 15:1–26, 1888). The list of irreducible configurations has been completed by Boben (Discret Math 307:331–344, 2007) in terms of their irreducible Levi graphs. In this paper we characterize irreducible pseudo 2-factor isomorphic cubic bipartite graphs proving that the only pseudo 2-factor isomorphic irreducible Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained characterization allows us to partially prove the above Conjecture.     

Intrinsically linked signed graphs in projective space

We define a signed embedding of a signed graph into real projective space to be an embedding such that an embedded cycle is 0-homologous if and only if it is balanced. We characterize signed graphs that have a linkless signed embedding. In particular, we exhibit 46 graphs that form the complete minor-minimal set of signed graphs that contain a non-split link for every signed embedding. With one trivial exception, these graphs are derived from different signings of the seven Petersen family graphs.     

Locally petersen graphs

A graph Γ is locally Petersen if, for each point t of Γ, the graph induced by Γ on all points adjacent to t is isomorphic to the Petersen graph. We prove that there are exactly three isomorphism classes of connected, locally Petersen graphs and further characterize these graphs by certain of their parameters.     

关于图的强符号全控制数

图的强符号全控制数有着许多重要的应用背景,因而确定其下界有重要的意义.本文提出了图的强符号全控制数的概念,在构造适当点集的基础上对其进行了研究,给出了：（1）一般图的强符号全控制数的5个独立可达的下界及达到其界值的图;（2）确定了圈、轮图、完全图、完全二部图的强符号全控制数的值.     

On natural isomorphisms of cycle permutation graphs

Two problems are approached in this paper. Given a permutation onn elements, which permutations onn elements yield cycle permutation graphs isomorphic to the cycle permutation graph yielded by the given permutation? And, given two cycle permutation graphs, are they isomorphic? Here the author deals only with natural isomorphisms, those isomorphisms which map the outer and inner cycles of one cycle permutation graph to the outer and inner cycles of another cycle permutation graph. A theorem is stated which then allows the construction of the set of permutations which yield cycle permutation graphs isomorphic to a given cycle permutation graph by a natural isomorphism. Another theorem is presented which finds the number of such permutations through the use of groups of symmetry of certain drawings of cycles in the plane. These drawings are also used to determine whether two given cycle permutation graphs are isomorphic by a natural isomorphism. These two methods are then illustrated by using them to solve the first problem, restricted to natural isomorphism, for a certain class of cycle permutation graphs.     

符号圈拼接图的零度

令$\eta(\Gamma)$和$c(\Gamma)$是符号图$\Gamma$的零度和基本圈数. 一个符号圈拼接图是指每个块都是圈的连通符号图. 本文证明了对任意符号拼接图$\eta(\Gamma)\le c(\Gamma)+1$成立, 并且刻画了等号成立的极图, 推广了王登银等人(2022)在简单圈拼接图上的结果. 此外, 我们证明了任意的符号拼接图$\eta(\Gamma)\neq c(\Gamma)$, 给出了满足$\eta(\Gamma)=c(\Gamma)-1$的符号拼接图的一些性质并刻画处$\eta(\Gamma)=c(\Gamma)-1$的二部符号拼接图.     

Graphs in which neighborhoods of vertices are isomorphic to the Hoffman-Singleton graph

Connected graphs are studied in which neighborhoods of vertices are isomorphic to the Hoffman-Singleton graph (i.e., the strongly regular graph with parameters (50,7,0,1)). It is proved that a distance-regular graph in which neighborhoods of vertices are isomorphic to the Hoffman-Singleton graph has µ = 2.     

Six signed Petersen graphs,and their automorphisms

Up to switching isomorphism, there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process, we develop new properties of signed graphs, especially of their switching automorphism groups.     

Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices

The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph.