Circuit decompositions of join‐covered graphs

In this paper, we focus our attention on join‐covered graphs, that is, ±1‐weighted graphs, without negative circuits, in which every edge lies in a zero‐weight circuit. Join covered graphs are a natural generalization of matching‐covered graphs. Many important properties of matching covered graphs, such as the existence of a canonical partition, tight cut decomposition and ear decomposition, have been generalized to join covered graphs by A. Seb? [5]. In this paper we prove that any two edges of a join‐covered graph lie on a zero‐weight circuit (under an equivalent weighting), generalize this statement to an arbitrary number of edges, and characterize minimal bipartite join‐covered graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 220–233, 2009     

Coincidence of the sets of minimal and irreducible join graphs over primary structure of algebraic Bayesian networks

An algebraic Bayesian network (ABN) is a probabilistic-logic graphical model of bases of knowledge patterns with uncertainty. A primary structure of an ABN is a set of knowledge patterns, that are ideals of conjunctions of positive literals except the empty conjunction endowed with scalar or interval probability estimates. A secondary ABN structure is represented by a graph constructed over the primary structure, which is called a join graph. From the point of view of learning of a global ABN structure, of interest are join graphs with the minimum number of edges and irreducible join graphs. A theorem on the coincidence of the sets of minimal and irreducible join graphs over the same primary structure is proved. A greedy algorithm constructing an arbitrary minimal join graph from a given primary structure is described. A theorem expressing the number of edges in a minimal join graph as the sum of the ranks of the incidence matrices of strong restrictions of a maximal join graph minus the number of significant weights is stated and proved. A generalized graph of maximal knowledge patterns (GGMKP) is a graph with the same vertex set as the join graph which is not subject to any constraints concerning the possibility of joining two vertices by an edge. It is proved that the pair consisting of the edge set of a maximal GGMKP and the set of all subsets of this graph such that the subtraction of any such subset from the maximal GGMKP yields an edge of the join graph on the same vertex set is a matroid.     

Chordal graphs and join spaces

A join space is an abstract model for partially ordered linear, spherical and projective geometries. A characterization for chordal graphs which are join spaces is given: a chordal graph is a join space if and only if it does not contain one of the two forbidden graphs as an induced subgraph.     

Ear decompositions of join covered graphs

Join covered graphs are ±1-weighted graphs, without negative circuits, in which every edge lies in a zero-weight circuit. Join covered graphs are a natural generalization of matching covered graphs. Many important properties of matching covered graphs have been generalized to join covered graphs. In this paper, we generalize Lovász and Plummerʼs ear decomposition theorem of matching covered graphs to join covered graphs.     

The join of split graphs whose half-strong endomorphisms form a monoid

In this paper, the half-strong endomorphisms of the join of split graphs are investigated. We give the conditions under which the half-strong endomorphisms of the join of split graphs form a monoid.     

The Join of Split Graphs Whose Regular Endomorphisms Form a Monoid

In this paper, the regular endomorphisms of the join of split graphs are investigated. We give a condition under which the regular endomorphisms of the join of split graphs form a monoid.     

Signed graphs with at most three eigenvalues

Czechoslovak Mathematical Journal - We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or...     

Total chromatic numbers

We determine the total chromatic number of the join of two paths, the cartesian product of two paths, the cartesian product of a path and a cycle, the corona of two graphs and the theta graphs.     

Topological efficiency under graph operations

In this paper we present explicit formulas for computing the topological efficiency index of the most important graph operations such as the Cartesian product, composition, corona, join and hierarchical product of two graphs. We apply our results to compute this distance-related invariant for some important classes of molecular graphs and nano-structures by specializing components of these graph operations.     

广义联图的正则性

讨论了两个图的广义联图的End-正则性,给出了当图X、Y的广义联图G（y1,…ym）End-正则时,图X也End-正则应满足的条件。     

On betweenness-uniform graphs

The betweenness centrality of a vertex of a graph is the fraction of shortest paths between all pairs of vertices passing through that vertex. In this paper, we study properties and constructions of graphs whose vertices have the same value of betweenness centrality (betweenness-uniform graphs); we show that this property holds for distanceregular graphs (which include strongly regular graphs) and various graphs obtained by graph cloning and local join operation. In addition, we show that, for sufficiently large n, there are superpolynomially many betweenness-uniform graphs on n vertices, and explore the structure of betweenness-uniform graphs having a universal or sub-universal vertex.     

On kernels in perfect graphs

A kernel of a digraphD is a set of vertices which is both independent and absorbant. In 1983, C. Berge and P. Duchet conjectured that an undirected graphG is perfect if and only if the following condition is fulfilled: ifD is an orientation ofG (where pairs of opposite arcs are allowed) and if every clique ofD has a kernel thenD has a kernel. We prove here the conjecture for the complements of strongly perfect graphs and establish that a minimal counterexample to the conjecture is not a complete join of an independent set with another graph.     

Graphs for which the least eigenvalue is minimal, I

Let G be a connected graph whose least eigenvalue λ(G) is minimal among the connected graphs of prescribed order and size. We show first that either G is complete or λ(G) is a simple eigenvalue. In the latter case, the sign pattern of a corresponding eigenvector determines a partition of the vertex set, and we study the structure of G in terms of this partition. We find that G is either bipartite or the join of two graphs of a simple form.     

一个五阶图与路、圈的联图的交叉数

目前已经确定的两个图的联图的交叉数结果较少.设H是由一个4圈及一个孤立点所构成的5阶图.研究了图H与路、圈的联图的交叉数,得到了cr(H+P_n)=Z(5,n)+[n/2]+l,cr(H+C_n):Z(5,n)+[n/2]+2,其中,P_n与C_n分别表示含n个顶点的路与圈.     

On the k-index of graphs

Given a graph, join every two vertices which are at a distance greater than a fixed integer k (>1)by a new path of length k. Thus a graph transformation is defined. The least number of iterations of this transformation such that the last iteration does not change the graph, is called the k-index of the original graph. In the present paper the graphs are classified according to their k-indices. The results are applied in the study of so-called tied graphs.     

Some Cohen–Macaulay and Unmixed Binomial Edge Ideals

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen–Macaulay and unmixed. So that we generalize the results of Ene, Herzog, and Hibi on block graphs. Moreover, we study unmixedness and Cohen–Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs.     

Spectra of graphs obtained by a generalization of the join graph operation

Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.     

Sufficient conditions for a graph to be edge-colorable with maximum degree colors

We study properties of graphs related to the existence of certain vertex and edge partitions. These properties give sufficient conditions for a graph to be Class 1 (i.e., edge-colorable with maximum degree colors). We apply these conditions for solving the classification problem for graphs with acyclic core (the subgraph induced by the maximum degree vertices), and for subclasses of join graphs and cobipartite graphs.     

Complement and Isomorphism on Bipolar Fuzzy Graphs

In this paper, we discuss some properties of the self complement and self weak complement bipolar fuzzy graphs, and get a sufficient condition for a bipolar fuzzy graph to be the self weak complement bipolar fuzzy graph. Also we investigate relations between operations union, join, and complement on bipolar fuzzy graphs.