Graph-reduction specifications (GRSs) are a powerful new method for specifying classes of pointer data structures (shapes). They cover important shapes, like various forms of balanced trees, that cannot be handled by existing methods. This report formally defines GRSs as graph-reduction systems with a signature restriction and an accepting graph. We are mainly interested in PGRSs, which are pol...

A 2-walk of a graph is a walk visiting every vertex at least once and at most twice. By generalizing decompositions of Tutte and Thomassen, Gao, Richter and Yu proved that every 3-connected planar graph contains a closed 2-walk such that all vertices visited twice are contained in 3separators. This seminal result generalizes Tutte’s theorem that every 4-connected planar graph is Hamiltonian as ...

It is shown that matrices over algebraically closed fields that are farthest apart in the commuting graph must be non-derogatory. Rank-one matrices and diagonalizable matrices are also characterized in terms of the commuting graph.

It is shown that matrices over algebraically closed fields that are farthest apart in the commuting graph must be non-derogatory. Rank-one matrices and diagonalizable matrices are also characterized in terms of the commuting graph.

‎in this paper, a new class of fuzzy sets called fuzzy strongly ${{g}^{*}}$-closed sets is introduced and its properties are investigated. moreover, we study some more properties of this type of closed spaces.

This work has two aims: First, we introduce a powerful technique for proving clique divergence when the graph satisfies a certain symmetry condition. Second, we prove that each closed surface admits a clique divergent triangulation. By definition, a graph is clique divergent if the orders of its iterated clique graphs tend to infinity, and the clique graph of a graph is the intersection graph o...

Let G be a simple graph with vertex set V(G) and edge set E(G). For each a in V(G), let N(a) = {x E V(G) : ax E E(G)} be the set of neighbours of a. A subset S of V(G) is called a closed set of G if, for each pair of distinct elements u, b in S, N(a) ON(b) s S. Let Z’(G) be the family of closed sets of G, inclusive of the empty set 0. Evidently, Z(G) is closed under arbitrary intersection and i...

Given a graph property P, it is interesting to determine the typical structure of graphs that satisfy P. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if P is a monotone property and r is the largest integer for which every r-colorable graph satis es P, then almost every gra...

Graph topologies for nonlinear operators which admit coprime factorisations are defined w.r.t. a gain function notion of stability in a general normed signal space setting. Several metrics are also defined and their relationship to the graph topologies are examined. In particular, relationships between nonlinear generalisations of the gap and graph metrics, Georgiou-type formulae and the graph ...