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Induced matching vs edge open packing: trees and product graphs
Get PDFGiven a graph , the maximum size of an induced subgraph of each component of which is a star is called the edge open packing number, , of . Similarly, the maximum size of an induced subgraph of each component of which is the star is the induced matching number, , of . While the inequality clearly holds for all graphs , we provide a structural characterization of those trees that attain the equality. We prove that the induced matching number of the lexicographic product of arbitrary two graphs and equals . By similar techniques, we prove sharp lower and upper bounds on the edge open packing number of the lexicographic product of graphs, which in particular lead to NP-hardness results in triangular graphs for both invariants studied in this paper. For the direct product of two graphs we provide lower bounds on and , both of which are widely sharp. We also present sharp lower bounds for both invariants in the Cartesian and the strong product of two graphs. Finally, we consider the edge open packing number in hypercubes establishing the exact values of when is a power of , and present a closed formula for the induced matching number of the rooted product of arbitrary two graphs over an arbitrary root vertex
Protein stacking on the APTES-functionalized pyrochlore ▫▫ clusters for ultrasensitive and selective immunosensing
Get PDFWith their unique physicochemical properties, such as metallic-like conductivity, favorable (electro)catalytic properties, electrochemical stability, and ease of functionalization, pyrochlores have found applications in various fields such as solid oxide fuel cells, batteries, thick film resistors, and temperature sensorshowever, there are no reports on their application in electrochemical immunosensing. In this study, we exploited the (electro)catalytic nature and stability of the pyrochlore Bi2Ru2O7 clusters silanized with (3-aminopropyl)triethoxysilane (APTES) to demonstrate their potential for the effective stacking of functional proteins. Characterization of the clusters by XPS disclosed a dual environment of Bi, also indicating the presence of Bi2O3 alongside APTES-Bi2Ru2O7 clusters and, importantly, the predominant involvement of pyrochlore moieties in subsequent protein stacking. After stacking protein A and antibodies, the immunosensor revealed a nearly interference-free operation, high sensitivity, a detection limit of 118 fM SARS-CoV-2 spike protein, and operation in a wide examined concentration range of 10−5−10−1 μg mL−1 with an r2 of 0.98. In combination with a short incubation time of 30 min, the pyrochlore-based immunosensor provides a solid platform for future point-of-need applications
