On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid of a lexicographic product G[H] of graphs G,H to be the wreath product of the monoids and . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G.     

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs $G$ and $H$. Let $H^k\left[G\right]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k\left[G\right]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k\left[G\right]$ and $H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10¹⁰⁰ ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.     

Cut‐elimination Theorems for Some Infinitary Modal Logics

In this article, a cut‐free system TLM_ω1 for infinitary propositional modal logic is proposed which is complete with respect to the class of all Kripke frames.The system TLM_ω1 is a kind of Gentzen style sequent calculus, but a sequent of TLM_ω1 is defined as a finite tree of sequents in a standard sense. We prove the cut‐elimination theorem for TLM_ω1 via its Kripke completeness.     

A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

In 1960, Sabidussi conjectured that if a graph G is isomorphic to the lexicographic product G[G], then the wreath product of by itself is a proper subgroup of . A positive answer is provided by constructing an automorphism Ψ of G[G] which satisfies: for every vertex x of G, there is an infinite subset I(x) of V(G) such that Ψ({x}×V(G))=I(x)×V(G).     

On the ultimate lexicographic Hall-ratio

The Hall-ratio ρ(G) of a graph G is the ratio of the number of vertices and the independence number maximized over all subgraphs of G. The ultimate lexicographic Hall-ratio of a graph G is defined as , where G^°n denotes the nth lexicographic power of G (that is, n times repeated substitution of G into itself). Here we prove the conjecture of Simonyi stating that the ultimate lexicographic Hall-ratio equals the fractional chromatic number for all graphs.     

The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G°H for a non-complete graph H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G°H)=2, as well as the lexicographic products T°H that enjoy g(T°H)=3g(G), when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G, a formula that expresses the exact geodetic number of G°H is established, where G is an arbitrary graph and H a non-complete graph.     

Decomposition tree of a lexicographic product of binary structures

Given a set S and a positive integer k, a binary structure is a function . The set S is denoted by V(B) and the integer k is denoted by . With each subset X of V(B) associate the binary substructure B[X] of B induced by X defined by B[X](x,y)=B(x,y) for any x≠y∈X. A subset X of V(B) is a clan of B if for any x,y∈X and v∈V(B)?X, B(x,v)=B(y,v) and B(v,x)=B(v,y). A subset X of V(B) is a hyperclan of B if X is a clan of B satisfying: for every clan Y of B, if X∩Y≠0?, then X⊆Y or Y⊆X. With each binary structure B associate the family Π(B) of the maximal proper and nonempty hyperclans under inclusion of B. The decomposition tree of a binary structure B is constituted by the hyperclans X of B such that Π(B[X])≠0? and by the elements of Π(B[X]). Given binary structures B and C such that , the lexicographic product B⌊C⌋ of C by B is defined on V(B)×V(C) as follows. For any (x,y)≠(x^′,y^′)∈V(B)×V(C), B⌊C⌋((x,x^′),(y,y^′))=B(x,y) if x≠y and B⌊C⌋((x,x^′),(y,y^′))=C(x^′,y^′) if x=y. The decomposition tree of the lexicographic product B⌊C⌋ is described from the decomposition trees of B and C.     

The fractional chromatic number, the Hall ratio, and the lexicographic product

Let χ_f denote the fractional chromatic number and ρ the Hall ratio, and let the lexicographic product of G and H be denoted GlexH. Main results: (i) ρ(GlexH)≤χ_f(G)ρ(H); (ii) if ρ(G)=χ_f(G) then ρ(GlexH)=ρ(G)ρ(H) for all H; (iii) χ_f−ρ is unbounded. In addition, the question of how big χ_f/ρ can be is discussed.     

On anti-magic labeling for graph products

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2,…,q} such that the vertex sums are pairwise distinct, where the vertex sum at one vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an anti-magic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K₂ are anti-magic. Recently, Alon et al. showed that this conjecture is true for dense graphs, i.e. it is true for p-vertex graphs with minimum degree Ω(logp). In this article, new classes of sparse anti-magic graphs are constructed through Cartesian products and lexicographic products.     

Quantifier elimination for the theory of algebraically closed valued fields with analytic structure

The theory of algebraically closed non‐Archimedean valued fields is proved to eliminate quantifiers in an analytic language similar to the one used by Cluckers, Lipshitz, and Robinson. The proof makes use of a uniform parameterized normalization theorem which is also proved in this paper. This theorem also has other consequences in the geometry of definable sets. The method of proving quantifier elimination in this paper for an analytic language does not require the algebraic quantifier elimination theorem of Weispfenning, unlike the customary method of proof used in similar earlier analytic quantifier elimination theorems. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogeneous factorisations of graph products

A homogeneous factorisation of a digraph Γ consists of a partition P={P₁,…,P_k} of the arc set AΓ and two vertex-transitive subgroups M?G?Aut(Γ) such that M fixes each P_i setwise while G leaves P invariant and permutes its parts transitively. Given two graphs Γ₁ and Γ₂ we consider several ways of taking a product of Γ₁ and Γ₂ to form a larger graph, namely the direct product, cartesian product and lexicographic product. We provide many constructions which enable us to lift homogeneous factorisations or certain arc partitions of Γ₁ and Γ₂, to homogeneous factorisations of the various products.     

The Gödel-McKinsey-Tarski embedding for infinitary intuitionistic logic and its extensions

The Gödel-McKinsey-Tarski embedding allows to view intuitionistic logic through the lenses of modal logic. In this work, an extension of the modal embedding to infinitary intuitionistic logic is introduced. First, a neighborhood semantics for a family of axiomatically presented infinitary modal logics is given and soundness and completeness are proved via the method of canonical models. The semantics is then exploited to obtain a labelled sequent calculus with good structural properties. Next, soundness and faithfulness of the embedding are established by transfinite induction on the height of derivations: the proof is obtained directly without resorting to non-constructive principles. Finally, the modal embedding is employed in order to relate classical, intuitionistic and modal derivability in infinitary logic extended with axioms.     

An approach to infinitary temporal proof theory

Aim of this work is to investigate from a proof-theoretic viewpoint a propositional and a predicate sequent calculus with an –type schema of inference that naturally interpret the propositional and the predicate until–free fragments of Linear Time Logic LTL respectively. The two calculi are based on a natural extension of ordinary sequents and of standard modal rules. We examine the pure propositional case (no extralogical axioms), the propositional and the first order predicate cases (both with a possibly infinite set of extralogical axioms). For each system we provide a syntactic proof of cut elimination and a proof of completeness.Supported by MIUR COFIN 02 Teoria dei Modelli e Teoria degli Insiemi, loro interazioni ed applicazioni.Supported by MIUR COFIN 02 PROTOCOLLO.Mathematics Subject Classification (2000):03B22, 03B45, 03F05     

Infinite products of filters

Stability of some classes of filters under the (infinite, Tikhonov) product operation is investigated. Applications to productivity of some types of set valued maps are given. Communicated by Ľubica Holá     

Dimensional results for Cartesian products of homogeneous Moran sets

M(J, {m _s * n _s }, {c _s }) be the collection of Cartesian products of two homogenous Moran sets with the same ratios {c _s } where J = [0, 1]×[0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {m _s n _s } or {c _s }.     

Equistable graphs, general partition graphs, triangle graphs, and graph products

In this paper we examine the connections between equistable graphs, general partition graphs and triangle graphs. While every general partition graph is equistable and every equistable graph is a triangle graph, not every triangle graph is equistable, and a conjecture due to Jim Orlin states that every equistable graph is a general partition graph. The conjecture holds within the class of chordal graphs; if true in general, it would provide a combinatorial characterization of equistable graphs.Exploiting the combinatorial features of triangle graphs and general partition graphs, we verify Orlin’s conjecture for several graph classes, including AT-free graphs and various product graphs. More specifically, we obtain a complete characterization of the equistable graphs that are non-prime with respect to the Cartesian or the tensor product, and provide some necessary and sufficient conditions for the equistability of strong, lexicographic and deleted lexicographic products. We also show that the general partition graphs are not closed under the strong product, answering a question by McAvaney et al.