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Let G be a graph with ∆(G) > 1. It can be shown that the domination number of the graph obtained from G by subdividing every edge exactly once is more than that of G. So, let ξ(G) be the least number of edges such that subdividing each of these edges exactly once results in a graph whose domination number is more than that of G. The parameter ξ(G) is called… (More)

Let G = (V, E) be a graph of order n and let D that for all v ∈ V, u∈ND(v) f (u) is a constant, called D-vertex magic constant. O'Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of… (More)

The infimum of the least eigenvalues of all finite induced subgraphs of an infinite graph is defined to be its least eigenvalue. the class of all finite graphs whose least eigenvalues −2 has been classified: (1) If a (finite) graph is connected and its least eigenvalue is at least −2, then either it is a generalized line graph or it is represented by the… (More)

A group in which every element is its own inverse is called a boolean group. We show that if G is an infinite boolean group, then there exist subsets A, B of G such that |A| = |B| = |G| and G \ {0} is the direct sum of A and B, i.e., for each (a, b) ∈ A × B, a + b = 0 and for each nonzero element g ∈ G, there exists a unique pair (a, b) ∈ A×B such that g =… (More)

An injective map from the vertex set of a graph G—its order may not be finite—to the set of all natural numbers is called an arithmetic (a geometric) labeling of G if the map from the edge set which assigns to each edge the sum (product) of the numbers assigned to its ends by the former map, is injective and the range of the latter map forms an arithmetic… (More)

In this note we prove that {0, 1, √ 2, √ 3, 2} is the set of all real numbers such that the following holds: every tree having an eigenvalue which is larger than has a subtree whose largest eigenvalue is. For terminology and notation, we follow [8]. The path with n vertices and the star with n edges are denoted by P n and K 1,n , respectively. The largest… (More)

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