• Kyungpook Mathematical Journal
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• v.61 no.3
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• pp.671-677
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• 2021

For a distance-regular graph with valency k, second largest eigenvalue r and diameter D, it is known that r ≥ $min\{\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2},\;a_3\}$ if D = 3 and r ≥ $\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2}$ if D ≥ 4, where λ = a₁. This result can be generalized to the class of edge-regular graphs. For an edge-regular graph with parameters (v, k, λ) and diameter D ≥ 4, we compare $\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2}$ with the local valency λ to find a relationship between the second largest eigenvalue and the local valency. For an edge-regular graph with diameter 3, we look at the number $\frac{{\lambda}-\bar{\mu}+\sqrt{({\lambda}-\bar{\mu})^2+4(k-\bar{\mu})}}{2}$, where $\bar{\mu}=\frac{k(k-1-{\lambda})}{v-k-1}$, and compare this number with the local valency λ to give a relationship between the second largest eigenvalue and the local valency. Also, we apply these relationships to distance-regular graphs.

• Economic and Environmental Geology
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• v.28 no.4
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• pp.351-367
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• 1995

The elemental geochemistry and oxygen isotopes of illite/smectite (I/S) have been studied in relationship to the mineralogical trend in the Reindeer D-27 well, Beaufort-Mackenzie Basin. The increase in concentrations of $K_2O$, Rb and rare earth elements (REE), the decrease in concentrations of tetrahedral elements such as Mg, Ti, Sc, Zn and Zr, and the increase in concentrations of tetrahedral elements such as Be and V can be related to I/S compositions that vary systematically with depth. Layer formulae of S- and I-layers are estimated as $[Al_{1.57}Fe_{.19}Mg_{.31}Ti_{.07}][Si_{3.84}Al_{.16}]O_{10}(OH)_2$ and $[Al_{1.84}Mg_{.16}][Si_{3.33}Al_{.67}]O_{10}(OH)_2$, respectively. The mobilization of REE appears to occur during illitization. The increase in concentrations of REE, especially La and Ce, with depth is probably linked to incorporation of ions with high valency (e.g. $V^{5+}$) in tetrahedral sites. The excess valency due to V is partly counter-balanced by ions with low valency (e.g. $Be^{2+}$) and, in turn, the local valency deficiency caused by $Be^{2+}$ could be compensated by high-charge interlayer cations such as REE (＋3). ${\delta}^{18}O$ values of I/S range from 2.91 to 15.72‰ (SMOW), and increase with depth, contrasting to trends observed in the Gulf Coast and elsewhere. The increase in ${\delta}^{18}O$ of I/S results from the rapid increase in ${\delta}^{18}O$ of pore water that overcomes the decrease in temperature-dependent fractionation values with increasing burial depth (${\delta}^{18}O_{pore\;water}>-d{\Delta}/_{I/S-water};\;d{\delta}^{18}O_{I/S}>0$). Calculated ${\delta}^{18}O$ values of pore water in equilibrium with I/S suggest that the original water was probably meteoric water. The stratification of pore water is postulated from the presence of an isotopically light interval, about 450m thick. The depth range of the isotopically light zone overlaps, but does not coincide with the interval of lowered I-content and $K_2O$ concentrations, suggesting that oxygens may have been exchanged independently of mineralogical and geochemical reactions.