Let G[F,Vk, Huv] be the graph with k pockets, where F is a simple graph of order n ≥ 1,Vk= {v1,v2,··· ,vk} is a subset of the vertex set of F and Hvis a simple graph of order m ≥ 2,v is a specified vertex of Hv. Also let G[F,Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n ≥ 2, Ek= {e1,e2,···ek} is a subset of the edge set of F and Huvis a simple graph of order m ≥ 3, uv is a specified edge of Huvsuch that Huv− u is isomorphic to Huv− v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F,Vk, Hv] and G[F,Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.

The dolomite dust-emulsified asphalt composite with excellent mechanical properties was successfully prepared using alkali activation. The effects of different alkali concentrations and emulsified asphalt contents on the mechanical properties of the materials were studied. The mechanical properties and microstructure of the composites were analyzed using compressive strength tests, bending strength tests, and SEM characterization. The experimental results show that the specimens have excellent mechanical properties: the 7-day compressive strength can reach 76.67 MPa, and the bending strength is about twice that of silicate-based geopolymer-emulsified asphalt composite. With an increase in emulsified asphalt content, the compressive strength of the samples decreases, while the bending strength increases first and then decreases. When the emulsified asphalt content is 1%, the bending strength of the sample is up to 28.81 MPa, which is 25% higher than that of the sample without emulsified asphalt. This indicates that an appropriate emulsified asphalt content can play a toughening role in the system, providing a new idea for designing high-toughness alkali-activated materials.