Table 4 Upper bounds of \(\pmb{R(G)}\) for bipartite graphs
From: Bounding the HL-index of a graph: a majorization approach
n | m | R ( G ) | Bound ( 17 ) | Bound ( 23 ) | Bound ( 4 ) | Bound ( 6 ) |
|---|---|---|---|---|---|---|
10 | 13 | 0.46 | 1.75 | 1.47 | 1.52 | 1.62 |
10 | 25 | 0.00 | 0.00 | 1.00 | 1.00 | 1.62 |
20 | 45 | 0.10 | 2.97 | 1.90 | 1.94 | 2.08 |
20 | 91 | 0.00 | 1.04 | 1.16 | 1.77 | 2.08 |
50 | 315 | 0.06 | 3.82 | 2.86 | 2.95 | 3.00 |
50 | 560 | 0.00 | 1.74 | 1.34 | 2.39 | 3.00 |
100 | 1,248 | 0.03 | 4.72 | 4.00 | 4.01 | 4.04 |
100 | 2,254 | 0.03 | 2.36 | 1.48 | 2.99 | 4.04 |
250 | 7,839 | 0.02 | 7.52 | 5.91 | 6.07 | 6.09 |
250 | 14,083 | 0.03 | 3.60 | 1.79 | 4.22 | 6.09 |
500 | 31,321 | 0.04 | 9.29 | 8.20 | 8.39 | 8.41 |
500 | 56,231 | 0.01 | 5.01 | 2.16 | 5.64 | 8.41 |
1,000 | 124,887 | 0.02 | 12.68 | 11.66 | 11.67 | 11.68 |
1,000 | 225,145 | 0.01 | 7.02 | 2.65 | 7.58 | 11.68 |