Wiener and Harary indices of Mycielskian graphs

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′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]. In this paper, we study the Wiener and Harary indices of the Mycielskian and closed Mycielskian graphs.