Wiener index for graphs and their line graphs with arbitrary large cyclomatic numbers

On the Reverse Wiener Indices of Unicyclic Graphs

“…Suppose that p = n − 4 and n ≥ 8. Then n−p−3 2 = 0, and thus (U n,4,4 ) = (U n, 3,4 ). Note that the number of pendent vertices of U n,4,4 = U n,4,4 (2, 0) is n − 5 and the number of pendent vertices of U n,3,4 = U n,3,4 (2, 1) is n − 4.…”

“…Its mathematical properties may be found in the surveys [4,5]. For recent results on Wiener index, see, e.g., [3,16,19]. Various graph invariants involving modification of the Wiener index have been developed, see, e.g., [1,7,11,13,15,17].…”

“…(G) ≤ n 2 − 2n + 7 with equality if and only if G = U n, 3,4 . (ii) If p = 3 and n = 7, then (G) ≤ 42 with equality if and only if G = U 7,3,4 or U k 7,4,4 (1, 1) for k = 0, 1.…”

On the Wiener Index of Some Total Graphs

“…It is a graph invariant much studied in both mathematical and chemical literature (see [12][13][14]17,21]). The concept of graph operator has found various applications in chemical research (see [8][9][10][11]16,18,19,22]). …”

Wiener Index of Graphs and Their Line Graphs

“…Minimal tricyclic graphs (λ(G) = 3) with property (1.1) have 12 vertices (71 graphs) [7]. Two infinite families of planar nonbipartite graphs of girth 3 or girth 4 with increasing cyclomatic number having property (1.1) are constructed in [9]. Dobrynin and Mel'nikov in [10,11] proved that for every λ(G) 3 there are planar bipartite graphs G of girth 4 with cyclomatic number λ(G) having property (1.1).…”