First order sentences about random graphs: Small number of alternations

Probabilities of Sentences about Very Sparse Random Graphs

W e consider random graphs with edge probability Pn-Q, where n is the number of vertices of the graph, /3 > 0 is fixed, and a = 1 or a = (I + 1)/1 for some fixed positive integer I . We prove that for every firstorder sentence, the probability that the sentence is true for the random graph has an asymptotic limit. Also, there is an eflective procedure for generating the value of the limit in cl...

Matching Integral Graphs of Small Order

In this paper, we study matching integral graphs of small order. A graph is called matching integral if the zeros of its matching polynomial are all integers. Matching integral graphs were first studied by Akbari, Khalashi, etc. They characterized all traceable graphs which are matching integral. They studied matching integral regular graphs. Furthermore, it has been shown that there is no matc...

Autoregressive Gaussian Random Vectors of First Order

Probabilities of First - Order Sentences about Unary Functions

Let / be any fixed positive integer and a a sentence in the first-order predicate calculus of/unary functions. For positive integers n, an «-structure is a model with universe {0,1,...,« 1} and/unary functions, and p(n,a) is the ratio of the number of «-structures satisfying a to n"f, the number of «-structures. We show that lim„_00/i(«, a) exists for all such a, and its value is given by an ex...

First-Order Definability of Trees and Sparse Random Graphs

Let D(G) be the smallest quantifier depth of a first order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first order logic. We show that almost surely D(G) = Θ( ln n ln ln n ), where G is a random tree of order n or the giant component of a random graph G(n, c n ) with constant c > 1. Thes...