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New classes of mean graphs
Abstract
A graph G = (V, E) with p vertices and q edges is said to be a mean graph if it is possible to label the vertices x ϵ V with distinct elements f (x) from 0, 1, 2, … , q in such a way that when each edge e = uv is labeled with if f (u) + f (v) is even and if f (u) + f (v) is odd, then the resulting edge labels are distinct. In this case f is called a mean labeling of G. In this paper we prove that Triangular Ladder TLn, TLn, ʘ K1, Tn, ʘ K1, D(Tn) ʘ K1, Qn ʘ K1, D(Qn) ʘ K1 are mean graphs.
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A graph G=(V,E) with p vertices and q edges is called a harmonic mean graph if it is possible to label the vertices x∈V with distinct labels f(x) from 1,2,...,q+1 in such a way that when each edge e=uv is labeled with f(uv)=2f(u)f(v) f(u)+f(v) or 2f(u)f(v) f(u)+f(v) then the edge labels are distinct. In this case f is called harmonic mean labeling of G. The concept of Harmonic mean labeling was introduced in [the authors, “Harmonic mean labelings of graphs” (to appear)]. In [loc. cit.] and in [Int. J. Contemp. Math. Sci. 7, No. 1–4, 197–208 (2012; Zbl 1251.05128)] we investigate the harmonic mean labeling of several standard graphs such as path, cycle comb, ladder, triangular snakes, quadrilateral snakes etc. In the present paper, we investigate the harmonic mean labeling for a polygonal chain, square of the path and dragon. Also we enumerate all harmonic mean graph of order ≤5.
A vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy a label depending on the vertex labels f(x) and f(y). The two best known labeling methods are called graceful and harmonious labelings. A function f is called a graceful labeling of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, . . . , q} such that, when each edge xy is assigned the label |f(x) - f(y)|, the resulting edge labels are distinct. A function f is called a harmonious labeling of a graph G with q edges if it is an injection from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x)+f(y) (mod q), the resulting edge labels are distinct. When G is a tree, exactly one label may be used on two vertices. Over the past three decades many variations of graceful and harmonious labelings have evolved and about 300 papers have been written on the subject of graph labeling....
A graph G=(V, E) with p vertices and q edges is said to be a mean graph if it is possible to label the vertices x?V with distinct elements f (x) from 0, 1, 2, ?, q in such a way that when each edge e=uv is labelled with (f(u)+f(v))/2 if f (u)+f (v) is even and (f (u)+f (v)+1)/2 if f (u)+f (v) is odd, then the resulting edge labels are distinct. f is called a mean labeling of G. In this paper, we investigate the mean labeling of caterpillar,
dragon, arbitrary super subdivision of a path and some graphs which are obtained from cycles and stars.
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