conclusions the clinical features are effective indicators for distinguishing ihh from hbl, and the spectral doppler may be a useful adjunct parameter for differential diagnosis. objectives to study the clinical manifestations and the ultrasound features of the pediatric patients for distinguishing ihh from hbl. patients and methods between 2002 and 2012, thirteen children with ihh and 38 child...

Let c be a proper edge colouring of a graph G = (V,E) with integers 1, 2, . . . , k. Then k ≥ ∆(G), while by Vizing’s theorem, no more than k = ∆(G)+ 1 is necessary for constructing such c. On the course of investigating irregularities in graphs, it has beenmoreover conjectured that only slightly larger k, i.e., k = ∆(G) + 2 enables enforcing additional strong feature of c, namely that it attri...

The i-local distinguishing number of G, denoted by LD i (G), was deened in 3]. Let T be a tree on n > 2 vertices with maximum degree bounded by some constant k. It is shown that LD 1 (T) = O(p n) and that for some trees this bound is tight. The result is extended to show that LD i (T) = O(n 1=(i+1)).

Given a graph G, a labeling c : V (G) → {1, 2, . . . , d} is said to be d-distinguishing if the only element in Aut(G) that preserves the labels is the identity. The distinguishing number of G, denoted by D(G), is the minimum d such that G has a d-distinguishing labeling. If G2H denotes the Cartesian product of G and H, let G 2 = G2G and G r = G2G r−1 . A graph G is said to be prime with respec...

We introduce the endomorphism distinguishing number De(G) of a graph G as the least cardinal d such that G has a vertex coloring with d colors that is only preserved by the trivial endomorphism. This generalizes the notion of the distinguishing number D(G) of a graph G, which is defined for automorphisms instead of endomorphisms. As the number of endomorphisms can vastly exceed the number of au...

We investigate the distinguishing index D′(G) of a graph G as the least number d such that G has an edge-colouring with d colours that is only preserved by the trivial automorphism. This is an analog to the notion of the distinguishing number D(G) of a graph G, which is defined for colourings of vertices. We obtain a general upper bound D′(G) ≤ ∆(G) unless G is a small cycle C3, C4 or C5. We al...