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For several decades, the disadvantages of traditional metallic rock and soil support structures have become increasingly prominent. As a result, it is urgent to seek lightweight, high-strength, durable and green economic substitute products made from the basalt fibre reinforced composites. Combining the two-phase material summing principle of composite material mechanics and the extension and application of Mohr–Coulomb strength criterion, the uniaxial tensile computing model of a cylindrical polymer specimen containing randomly distributed and discontinuous basalt fibres was established in accordance with the numerical simulation software Flac3D. Furthermore, the influence laws and intrinsic mechanism of the number of discrete fibres and different fibre distribution angles on the integral axial tensile properties of a cylindrical specimen with chopped basalt fibre thermosetting resin matrix composites were investigated and elucidated, and the stress–strain curve of basalt fibre thermosetting resin matrix composites were obtained. Consequently, the results have indicated that: i) the axial tensile strength of the polymer system can be enhanced through the incorporation of a moderate quantity of chopped fibres, with the resultant improvement typically not surpassing a 10% increment; ii) when the number of fibres has increased to the quantitative state, namely, the fibre density reaches 1.02 pieces/cm3or more, the amplification of axial tensile strength of composite material will be controlled; iii) the fibre distribution directions, are maintaining a 0° parallel state to the axial direction of the specimen and the ones of load application, will ensure that the tensile strength of the composite can be sufficiently utilized. As the production process within the composite industry have become increasingly adept, the research results hold substantial value for further promoting the development, testing and optimizing application of various types of high-performance basalt fibre reinforced composite products.

A path in an edge-colored (respectively vertex-colored) graph Gis rainbow(respectively vertex-rainbow) if no two edges (respectively internal vertices) of the path are colored the same. An edge-colored (respectively vertex-colored) graph Gis rainbow connected(respectively vertex-rainbow connected) if every two distinct vertices are connected by a rainbow (respectively vertex-rainbow) path. The rainbow connection number rc(G) (respectively vertex-rainbow connection number rvc(G)) of Gis the smallest number of colors that are needed in order to make Grainbow connected (respectively vertex-rainbow connected). In this paper, we show that for a connected graph Gand any edge e= xy∈ E(G), rvc(G) ≤ rvc(G− e) ≤ rvc(G) + dG−e(x, y) − 1 if G− eis connected. For any two connected, non-trivial graphs Gand H, rad(G□H)−1 ≤ rvc(G□H) ≤ 2rad(G□H), where G□His the Cartesian product of Gand H. For any two non-trivial graphs Gand Hsuch that Gis connected, rvc(G◦ H) = 1 if diam(G◦ H) ≤ 2, rad(G) − 1 ≤ rvc(G◦ H) ≤ 2rad(G) if diam(G) > 2, where G◦ His the lexicographic product of Gand H. For the line graph L(G) of a graph Gwe show that rvc(L(G)) ≤ rc(G), which is the first known nontrivial inequality between the rainbow connection number and vertex-rainbow connection number. Moreover, the bounds reported are tight or tight up to additive constants.

A connected edge dominating setof a connected graph G=(V,E)is a subset Xof Esuch that the edge-induced subgraph G[X] is connected and each e∈E(G)\Xhas at least one neighbor in X. The connected edge domination numberγc′(G)of Gis the minimum cardinality of a connected edge dominating set of G. An edge dominating set Xof a graph Gis called a 2-edge-connected edge dominating setif G[X] is 2-edge-connected. The 2-edge-connected edge dominating numberγ2ec′(G)is the minimum size of a 2-edge-connected edge dominating set of G. In this paper, we obtain the sharp lower bounds for γc′(G)+γc′(G¯)and γc′(G)·γc′(G¯). Moreover, we characterize the classes of graphs attaining the lower bounds and study the relationship between γc′(G)and several other parameters, such as independent number and vertex cover number. In addition, we show that 3≤γ2ec′(G)≤⌊32(n-1)⌋if Gis 2-edge-connected. We also obtain the upper and lower bounds for γ2ec′(G)+γ2ec′(G¯)and γ2ec′(G)·γ2ec′(G¯)and characterize the classes of graphs attaining the lower bounds.

The k-forcing number of a graph G, denoted by $$F_k(G)$$ Fk(G) , was introduced by Amos et al. It is a generalization of the zero forcing number of a graph G, denoted by Z(G). Amos et al. proved that for a connected graph Gof order nwith maximum degree $$\varDelta \ge 2$$ Δ≥2 , $$Z(G)=F_1(G)\le \frac{(\varDelta -2)n+2}{\varDelta -1}$$ Z(G)=F1(G)≤(Δ-2)n+2Δ-1 , and this inequality is sharp. Moreover, they posed a conjecture that $$Z(G)=F_1(G)=\frac{(\varDelta -2)n+2}{\varDelta -1}$$ Z(G)=F1(G)=(Δ-2)n+2Δ-1 if and only if $$G=C_n$$ G=Cn , $$G=K_{\varDelta +1}$$ G=KΔ+1 or $$G=K_{\varDelta ,\varDelta }$$ G=KΔ,Δ . In this paper, we prove that this conjecture is true. Moreover, we point out a mistake in their paper and get a stronger result which shows that $$F_{n-1}(G)=1$$ Fn-1(G)=1 if and only if Gis connected and $$F_k(G)=n-k$$ Fk(G)=n-k if and only if $$G=K_n$$ G=Kn for $$k\le n-2$$ k≤n-2 .

Let G= (V,E) be a graph and S⊆ V. We say that Sis a dominating set of G, if each vertex in V\ Shas a neighbor in S. Moreover, we say that Sis a connected (respectively, 2-edge connected or 2-connected) dominating set of Gif G[S] is connected (respectively, 2-edge connected or 2-connected). The domination (respectively, connected domination, or 2- edge connected domination, or 2-connected domination) number of Gis the cardinality of a minimum dominating (respectively, connected dominating, or 2-edge connected dominating, or 2-connected dominating) set of G, and is denoted γ(G) (respectively γ1(G), or γ′ 2(G), or γ2(G)). A well-known result of Duchet and Meyniel states that γ1(G) ≤ 3γ(G) − 2 for any connected graph G. We show that if γ(G) ≥ 2, then γ′ 2(G) ≤ 5γ(G) − 4 when Gis a 2-edge connected graph and γ2(G) ≤ 11γ(G) − 13 when Gis a 2-connected triangle-free graph.

A path in an edge-colored graph Gis rainbowif no two edges of the path are colored the same. The rainbow connection number rc(G) of Gis the smallest integer kfor which there exists a k-edge-coloring of Gsuch that every pair of distinct vertices of Gis connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph Gwith diameter d. In this paper, we shall show that 7 ≤ f(3) ≤ 9.