Leonid S. Melnikov

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The Wiener number, W (G), is the sum of the distances of all pairs of vertices in a graph G. Infinite families of graphs with increasing cyclomatic number and the property W (G) = W (L(G)) are presented, where L(G) denotes the line graph of G. This gives a positive (partial) answer to an open question posed in an earlier paper by Gutman, Jovašević, and(More)
Let G be a 4-regular planar graph and suppose that G has a cycle decomposition S (i.e., each edge of G is in exactly one cycle of the decomposition) with every pair of adjacent edges on a face always in different cycles of S. Such graphs, called Grötzsch–Sachs graphs, arise as a superposition of simple closed curves in the plane with tangencies disallowed.(More)
Given a set of integers S; G(S) = (S; E) is a graph, where the edge uv exists if and only if u+ v∈ S. A graph G = (V; E) is an integral sum graph or ISG if there exists a set S ⊂ Z such that G=G(S). This set is called a labeling of G. The main results of this paper concern regular ISGs. It is proved that all 2-regular graphs with the exception of C4 are(More)