the order graph of a group $g$, denoted by $gamma^*(g)$, is a graph whose vertices are subgroups of $g$ and two distinct vertices $h$ and $k$ are adjacent if and only if $|h|big{|}|k|$ or $|k|big{|}|h|$. in this paper, we study the connectivity and diameter of this  graph. also we give a relation between the order graph and prime  graph of a group.

let f, g and h be non-empty graphs. the notation f → (g,h) means that if any edge of f is colored by red or blue, then either the red subgraph of f con- tains a graph g or the blue subgraph of f contains a graph h. a graph f (without isolated vertices) is called a ramsey (g,h)−minimal if f → (g,h) and for every e ∈ e(f), (f − e) 9 (g,h). the set of all ramsey (g,h)−minimal graphs is denoted by ...

in this paper, we first collect the earlier results about some graph operations and then wepresent applications of these results in working with chemical graphs.

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Fräıssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups, and let m be a number t...

We study a natural hierarchy in first-order logic, namely the quantifier structure hierarchy, which gives a systematic classification of first-order formulas based on structural quantifier resource. We define a variant of Ehrenfeucht-Fräıssé games that characterizes quantifier classes and use it to prove that this hierarchy is strict over finite structures, using strategy compositions. Moreover...

We present a new proof technique for collapse results for first–order queries on databases which are embedded in N or R>0. Our proofs are by means of an explicitly constructed winning strategy for Duplicator in an Ehrenfeucht–Fräıssé game, and can deal with certain infinite databases where previous, highly involved methods fail. Our main result is that first–order logic has the natural–generic ...

We propose a refinement of the usual Ehrenfeucht-Fräıssé game. The new game will help us make finer distinctions than the traditional one. In particular, it can be used to measure not only quantifier rank but also lengths of conjunctions and disjunctions needed for expressing a given property. Our game is similar to the game in [1] and in [5]. The most common measure of complexity of a first or...

We show that for each n and m, there is an existential first order sentence which is NOT logically equivalent to a sentence of quantifier rank at most m in infinitary logic augmented with all generalized quantifiers of arity at most n. We use this to show the strictness of the quantifier rank hierarchies for various logics ranging from existential (or universal) fragments of first order logic t...

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