Table 2 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 |
|---|---|
\(d(1,v_{q}),~1\le q\le 63\) | 1 for \(q=27\). ; \(\frac{1}{2}\) for \(q=21\). ; \(\frac{1}{3}\) for \(q=2,9,13\). |
\(\frac{1}{4}\) for \(q=5\). ; \(\frac{1}{5}\) for \(q=4,7,8,10,11,18,20,22,23,29,35,42,45,48,54,55,61\). | |
\(\frac{1}{6}\) for \(q=3,12,14,15,16,25,26,28,32,38\). | |
\(\frac{1}{7}\) for \(q=19,24,30,49,56\). and 0 otherwise | |
\(d(2,v_{q}),~1\le q\le 63\) | 1 for \(q=1,7,8,19,20,49\). |
\(\frac{1}{2}\) for \(q=3,4,9,10,11,12,14,16,27,28,29,32,35,38,42,45,48,54,61\). | |
\(\frac{1}{3}\) for \(q=13,15,21,22,23,24,25,26,30,55,56\). | |
\(\frac{1}{4}\) for \(q=5\). | |
\(\frac{1}{5}\) for \(q=18\). and 0 otherwise | |
\(d(3,v_{q}),~1\le q\le 63\) | 1 for \(q=1,7,8,12,21,22,23,25,26,32,42,45,54,55\) ; \(\frac{1}{2}\) for \(q=4,5,13,27\). |
\(\frac{1}{3}\) for \(q=2,9,10,11,15,18,19,20,24,35,38,48,49,56,61\). | |
\(\frac{1}{4}\) for \(q=14,16,28,29\). | |
\(\frac{1}{5}\) for \(q=30.\) and 0 otherwise | |
\(d(4,v_{q}),~1\le q\le 63\) | 1 for \(q=7,8,15,24,30,38,56\). ; \(\frac{1}{2}\) for \(q=1,2,3,9\). |
\(\frac{1}{3}\) for \(q=19,20,21,22,23,25,26,27,32,42,45,49\). | |
\(\frac{1}{4}\) for \(q=5,10,13,14,16,28,29,35,48,55,61.\) | |
\(\frac{1}{5}\) for \(q=11,12,18.\) and 0 otherwise | |
\(d(5,v_{q}),~1\le q\le 63\) | 1 for \(q=2,4,7,10,11,18,20,22,23,29,35,42,45,48,54,55,61\). |
\(\frac{1}{2}\) for \(q=1,3,9,12,14,15,16,21,25,26,28,32,38\). | |
\(\frac{1}{3}\) for \(q=19,24,27,30,49,56.\) | |
\(\frac{1}{6}\) for \(q=13\). and 0 otherwise | |
\(d(6,v_{q}),~1\le q\le 63\) | 1 for \(q=3,5,9,10,16,21,23,25,26,28,29,32,35,38,42,45,48,61\). |
\(\frac{1}{2}\) for \(q=1,2,4,7,8,11,12,18,20,22,54,55\). | |
\(\frac{1}{3}\) for \(q=14\). | |
\(\frac{1}{4}\) for \(q=13,15,19,24,27,30,49,56\). and 0 otherwise | |
\(d(7,v_{q}),~1\le q\le 63\) | 0 for \(1\le q\le 63\) |
\(d(8,v_{q}),~1\le q\le 63\) | 1 for \(q=10\). ; \(\frac{1}{2}\) for \(q=1,2,7,9.\) ; \(\frac{1}{3}\) for \(q=19,20,27,49.\) |
\(\frac{1}{4}\) for \(q=3,4,11,12,14,16,28,29,32,35,38,42,45,48,54,61\). | |
\(\frac{1}{5}\) for \(q=13,15,21,22,23,24,25,26,30,55,56\). | |
\(\frac{1}{6}\) for \(q=5\). ; \(\frac{1}{7}\) for \(q=18\). | |
and 0 otherwise | |
\(d(9,v_{q}),~1\le q\le 63\) | 1 for \(q=7,8\). ; \(\frac{1}{2}\) 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 \(q=13,15,21,22,23,24,25,26,30,55,56\). | |
\(\frac{1}{7}\) for \(q=5,\). ; \(\frac{1}{8}\) for \(q=18\). and 0 otherwise | |
\(d(10,v_{q}),~1\le q\le 63\) | 1 for \(q=1,2,7,9\). ; \(\frac{1}{2}\) for \(q=8,19,20,27,49\). |
\(\frac{1}{3}\) 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 | |
\(d(11,v_{q}),~1\le q\le 63\) | 0 for \(1\le q\le 63\) |
\(d(12,v_{q}),~1\le q\le 63\) | \(\frac{1}{2}\) for \(q=9\). ; \(\frac{1}{3}\) for \(q=7,8\). ; \(\frac{1}{4}\) for \(q=10.\) ; \(\frac{1}{5}\) for \(q=1,2.\) |
\(\frac{1}{6}\) for \(q=19,20,27,49\). ; \(\frac{1}{7}\) for \(q=3,4,11,14,16,28,29,32,35,38,42,45,48,54,61.\) | |
\(\frac{1}{8}\) for \(q=13,15,21,22,23,24,25,26,30,55,56\). | |
\(\frac{1}{9}\) for \(q=5\). | |
\(\frac{1}{10}\) for \(q=18\). and 0 otherwise |
