WIENER AND HARARY INDICES OF MYCIELSKIAN GRAPHS

WIENER AND HARARY INDICES OF MYCIELSKIAN GRAPHS

S. Goyal, Tanya

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Abstract

Let G = (V (G),E(G)) be a graph, where V = {v1, v2, . . . vn}. Let V ′ = {v′ 1, v′ 2 , . . . , v′ n } be the twin of the vertex set V (G). The Mycielskian graph M(G) of G is defined as the graph whose vertex set is V (G) ∪ V ′(G) ∪ {w} and the edge set is E(G) ∪ {viv′ j : vivj ∈ E(G)} ∪ {v′ iw ∈ V ′(G)}. The vertex v′ i is the twin of the vertex vi (or vi is twin of the vertex v′ i) and the vertex w is the root of M(G). The closed Mycielskian graph M[G] of G is defined as the graph whose vertex set is V (G) ∪ V ′(G) ∪ {w} and the edge set is E(G)∪{viv′ j : vivj ∈ E(G)}∪{viv′ i : i = 1, 2, . . . , n}∪{v′ i w ∈ V ′(G)}. The vertex v′ i is the twin of the vertex vi (or vi is twin of the vertex v′ i ) and the vertex w is the root of M[G]. In this paper, we study the Wiener and Harary indices of the Mycielskian and closed Mycielskian graphs.

Keywords

closed splitting graph, shadow graph, closed shadow graph, Mycielskian graph, closed Mycielskian graph.