… In other words a graph G* is an H-extension of a given graph G if G* is the resulting
graph of a sequence of immediate H-extensions whose initial graph is G. When thc section …

… We consider rooted subgraphs in random graphs, that is, extension counts such as (i) the
number of triangles containing a given vertex or (ii) the number of paths of length three …

… G; H /-extensions of the subgraph in G induced on fQx1;:::; Qxkg; we say that the graph X is
the subgraph of the graph Y induced on a set SV .Y / if VX/ DS and for any vertices x; y 2 S the …

… The rooted graph is denoted by the pair (R, H). For our purposes rooted graphs are a convenient
… the almost always character of random graphs. But we also feel that the theory of rooted …

… Further in the work, in many places we restrict the probability space of graphs to only those
graphs in which this property is satisfied for all \(\ell \leq k\) , where \(k\) is a predefined fixed …

… G satisfies the property L ” is both an element of F N and a set of all graphs in Ω N , that satisfy
the property L (we write G ⊨ L ). The random graph satisfies the property L almost surely if …

… This paper addresses the asymptotic behavior of probabilities of properties of random graphs
… that a graph has property L meaning by this the ele ment N that is the set of all graphs from …

… The random variable maps a graph G from Ω N to the number of all ( G , H ) -extensions of
G | { x 1 , … , x k } in G . A graph X is called a subgraph of a graph Y induced on a set S ⊂ V ( …

We show that the deficiency indices of the minimal Gaffney Laplacian on an infinite locally
finite metric graph are equal to the number of finite volume graph ends. Moreover, we provide …

… to be very useful in the study of random graphs. We begin with a general (though hardly the
… We do have is = 0(1) and L1 = 0(1) in our graph example so that the probability x does not …