Table 3 Shortest reciprocals path distance between vertices \(v_p\) and \(v_q\).
From: Harary index of the zero divisor graph of upper triangular matrices
Vertex pairs | Shortest reciprocals path distance between vertices |
|---|---|
\(d(13,v_{q}),~1\le q\le 63\) | 0 for \(1\le q\le 63\) |
\(d(14,v_{q}),~1\le q\le 63\) | \(\frac{1}{2}\) for \(q=2\). ; \(\frac{1}{3}\) for \(q=1,3,7,8,12,19,20,21,22,23,25,26,32,42,45,49,54,55.\) |
\(\frac{1}{4}\) for \(q=4,8,10,11,13,16,27,28,29,35,38,48,61.\) | |
\(\frac{1}{5}\) for \(q=15,18,24,30,56.\) and 0 otherwise | |
\(d(15,v_{q}),~1\le q\le 63\) | 1 for \(q=8,9\). ; \(\frac{1}{2}\) for \(q=7,10\). ; \(\frac{1}{3}\) for \(q=1,2\). ; \(\frac{1}{4}\) for \(q=19,20,27,49\). |
\(\frac{1}{5}\) for \(q=3,4,11,12,14,16,28,29,32,35,38,42,45,48,54,61\). | |
\(\frac{1}{6}\) for \(q=13,21,22,23,24,25,26,30,55,56\). | |
\(\frac{1}{7}\) for \(q=5\). and 0 otherwise | |
\(d(16,v_{q}),~1\le q\le 63\) | 1 for \(q=8,9\). ; \(\frac{1}{2}\) for \(q=7,10\). |
\(\frac{1}{3}\) for \(q=1,2\). ; \(\frac{1}{4}\) for \(q=15,19,20,27,49\). | |
\(\frac{1}{5}\) for \(q=3,4,11,12,14,28,29,32,35,38,42,45,48,54,61\). | |
\(\frac{1}{6}\) for \(q=13,21,22,23,24,25,26,30,55,56\). | |
\(\frac{1}{7}\) for \(q=5\). and 0 otherwise | |
\(d(17,v_{q}),~1\le q\le 63\) | 1 for \(q=2,4,7,8,9,10,11,12,14,16,20,21,22,23,28,29,32,35,38,42\) |
, 45, 48, 54, 55, 61. ; \(\frac{1}{2}\) for \(q=1,3,13,15,24,25,26,27,30\). | |
\(\frac{1}{3}\) for \(q=18,19,49,56\). ; \(\frac{1}{4}\) for \(q=5\). and 0 otherwise | |
\(d(18,v_{q}),~1\le q\le 63\) | 1 for \(q=1,3,7,8,9,10,11,12,16,21,22,23,25,26,28,29,32,35,38,42,45\) |
, 48, 54, 55, 61. | |
\(\frac{1}{2}\) for \(q=2,4,5,13\). ; \(\frac{1}{3}\) for \(q=20\). | |
\(\frac{1}{5}\) for \(q=14\). | |
\(\frac{1}{6}\) for \(q=15,24,27,30.\) and 0 otherwise | |
\(d(19,v_{q}),~1\le q\le 63\) | 1 for \(q=7,8,9\). ; \(\frac{1}{2}\) for \(q=10\). ; \(\frac{1}{3}\) for \(q=1,2\). ; \(\frac{1}{4}\) for \(q=19,20,27,49\). |
\(\frac{1}{5}\) for \(q=3,4,11,12,14,16,28,29,32,35,38,42,45,48,54,61\). | |
\(\frac{1}{6}\) for \(q=13,21,22,23,24,25,26,30,55,56\). | |
\(\frac{1}{7}\) for \(q=5\). and 0 otherwise | |
\(d(20,v_{q}),~1\le q\le 63\) | 1 for \(q=3,4,7,8,9,10,11,12,14,16,21,22,23,25,26,28,29,32,35,38\) |
, 42, 45, 54, 55, 61. ; \(\frac{1}{2}\) for \(q=2,13,15,24,27,30,48\). | |
\(\frac{1}{3}\) for \(q=1,5,19\). | |
\(\frac{1}{4}\) for \(q=18.\) and 0 otherwise | |
\(d(21,v_{q}),~1\le q\le 63\) | 1 for \(q=2,5,9,13\). |
\(\frac{1}{2}\) for \(q=4,7,8,10,11,18,19,20,22,23,29,35,42,45,48,54,55,61.\) | |
\(\frac{1}{3}\) for \(q=1,3,12,26,28,32,38\). | |
\(\frac{1}{4}\) for \(q=14,15,16,24,27,30,49,56.\) and 0 otherwise | |
\(d(22,v_{q}),~1\le q\le 63\) | 0 for \(1\le q\le 63.\) |
\(d(23,v_{q}),~1\le q\le 63\) | 0 for \(1\le q\le 63\) |
\(d(24,v_{q}),~1\le q\le 63\) | 0 for \(1\le q\le 63\) |
\(d(25,v_{q}),~1\le q\le 63\) | 1 for \(q=5,7\). ; \(\frac{1}{2}\) for \(q=2,4,10,11,18,20,22,23,29,35,42,45,48,54,55,61\). |
\(\frac{1}{3}\) for \(q=1,3,9,12,14,15,16,21,25,26,28,32,38\). | |
\(\frac{1}{4}\) for \(q=19,24,27,30,49,56.\) | |
\(\frac{1}{7}\) for \(q=13\). and 0 otherwise | |
\(d(26,v_{q}),~1\le q\le 63\) | 1 for \(q=4,15,21\). ; \(\frac{1}{2}\) for \(q=2,5,7,8,9,13,24,30,38,56.\) |
\(\frac{1}{3}\) for \(q=1,3,11,18,19,20,22,23,29,35,42,45,48,54,55,61\). | |
\(\frac{1}{4}\) for \(q=12,25,27,28,32,49\). | |
\(\frac{1}{5}\) for \(q=14,16.\) and 0 otherwise | |
\(d(27,v_{q}),~1\le q\le 63\) | 1 for \(q=21\); \(\frac{1}{2}\) for \(q=2,5,9,13\). |
\(\frac{1}{3}\) for \(q=4,7,8,10,11,18,19,20,22,23,29,35,42,45,48,54,55,61.\) | |
\(\frac{1}{4}\) for \(q=1,3,12,26,28,32,38\). | |
\(\frac{1}{5}\) for \(q=14,15,16,24,30,49,56.\) and 0 otherwise | |
\(d(28,v_{q}),~1\le q\le 63\) | 1 for \(q=1,2,7,8,9\). ; \(\frac{1}{2}\) for \(q=10,19,20,27,49\). |
\(\frac{1}{3}\) for \(q=3,4,11,12,14,16,28,29,32,35,42,45,48,54,61\). | |
\(\frac{1}{4}\) for \(q=13,15,21,22,23,24,25,26,30,55,56\). | |
\(\frac{1}{5}\) for \(q=5\). | |
\(\frac{1}{6}\) for \(q=18\). and 0 otherwise |
