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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/cm3 or 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. [ABSTRACT FROM AUTHOR]

A path in an edge-colored (respectively vertex-colored) graph G is 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 G is 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 G is the smallest number of colors that are needed in order to make G rainbow connected (respectively vertex-rainbow connected). In this paper, we show that for a connected graph G and any edge e = xy ∈ E(G), rvc(G) ≤ rvc(G − e) ≤ rvc(G) + dG−e(x, y) − 1 if G − e is connected. For any two connected, non-trivial graphs G and H, rad(G□H)−1 ≤ rvc(G□H) ≤ 2rad(G□H), where G□H is the Cartesian product of G and H. For any two non-trivial graphs G and H such that G is connected, rvc(G ◦ H) = 1 if diam(G ◦ H) ≤ 2, rad(G) − 1 ≤ rvc(G ◦ H) ≤ 2rad(G) if diam(G) > 2, where G ◦ H is the lexicographic product of G and H. For the line graph L(G) of a graph G we 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.

In this paper, we focus on examining the K 1 , 2 -structure-connectivity of any connected graph. Let G be a connected graph with n vertices, we show that κ (G ; K 1 , 2) is well defined if diam (G) ≥ 4 , or n ≡ 1 (mod 3) , or G ∉ { C 5 , K n } when n ≡ 2 (mod 3) , or there exist three vertices u, v, w such that N G (u) ∩ (N G ({ v , w }) ∪ { v , w }) = ∅ when n ≡ 0 (mod 3) . Furthermore, if G has K 1 , 2 -structure-cut, we prove κ (G) / 3 ≤ κ (G ; K 1 , 2) ≤ κ (G) . [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a graph and S ⊆ V. We say that S is a dominating set of G, if each vertex in V \ S has a neighbor in S. Moreover, we say that S is a connected (respectively, 2-edge connected or 2-connected) dominating set of G if 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 G is 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 G is a 2-edge connected graph and γ2(G) ≤ 11γ(G) − 13 when G is a 2-connected triangle-free graph.

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

Given a connected graph G and S ⊆ V (G) with | S | ≥ 2 , an S -tree is a such subgraph T = (V ′ , E ′) of G that is a tree with S ⊆ V ′ . Two S -trees T and T ′ are edge-disjoint if E (T) ∩ E (T ′) = ∅. Let λ G (S) be the maximum size of a set of edge-disjoint S -trees in G. The λ k -connectivity of G is defined as λ k (G) = min { λ G (S) : S ⊆ V (G) , | S | = k }. In this paper, we first show some structural properties of edge-disjoint S -trees by Fan Lemma and König-ore Formula. Then, the λ 4 -connectivity of the Cartesian product of trees is determined. That is, let T n 1 , T n 2 , ... , T n k be trees, then λ 4 (T n 1 □ T n 1 ⋯ □ T n k ) = k if | V (T n i ) | ≥ 4 for each i ∈ { 1 , 2 , ... , k } , otherwise λ 4 (T n 1 □ T n 2 □ ⋯ □ T n k ) = k − 1. As corollaries, λ 4 -connectivity for some graph classes such as hypercubes and meshes can be obtained directly. [ABSTRACT FROM AUTHOR]

During the underground mining of coal resources, the rock pressure emerges acutely and the mine geological disasters occur frequently. It is of great significance to grasp the manifestation law of rock pressure in time to guide the safety production and operation in the pit. In this research, the calculation equations and concept of information entropy based on zone safety degree are primarily defined, and the synergetic theory of the maximum information entropy principle is combined to put forward the synergy prediction model of information entropy based on zone safety degree. In the meantime, the synergy prediction model of information entropy based on zone safety degree is employed to calculate and predict the first weighting step and the periodic weighting step of the main roof of the 9203 working face of Hengsheng Coal Mine in China's Shanxi Province, as well as verifying the validity and reliability of the synergy prediction model of information entropy based on zone safety degree by the comparison of similar simulation test results, which has presented a scientific basis for the effective control of rock pressure and roof management. [ABSTRACT FROM AUTHOR]

In order to conduct an intelligent early warning assessment of stope rock burst disasters in mining areas, and effectively prevent and control them, the principal component analysis (PCA) method was embraced to perform dimensionality reduction and feature information extraction from 10 main factors that affect the occurrence of rock bursts. On this basis, six principal component elements of the influencing factors of rock bursts have been obtained as the input vectors for an extreme learning machine (ELM). In the meantime, the parameter optimization ability of the PSO algorithm was adopted, the input weight values of the ELM and the threshold values of the hidden layer were optimized, and the functions of the three models were completely combined. Therefore, an early warning model of rock bursts based on the PCA-PSO-ELM combined algorithm was creatively proposed and the risk rank of rock bursts in the Yanshitai Coal Mine was predicted and evaluated. Consequently, the research results indicated that the prediction accuracy of the PCA-PSO-ELM model improved the prediction performance and generalization ability and reached a 100% contrast with the three models, namely the BP neural network, the radial basis function, and the extreme learning machine, which presented an updated method for the early warning investigation of rock burst disasters and had favorable engineering significance. [ABSTRACT FROM AUTHOR]

It is generally acknowledged that the stability evaluation of surrounding rock denotes nonlinear complex system engineering. In order to accurately and quantitatively assess the safety states of surrounding rock and provide a scientific basis for the prevention and control of surrounding rock stability, the analysis method of the synergetic theory of information entropy using the failure approach index has been proposed. By means of deriving the general relationship between the total two-dimensional plastic shear strain and the total three-dimensional plastic shear strain and obtaining the numerical limit analysis step of the plastic shear strain, the threshold value of the ultimate plastic shear strain can be determined, which has provided the key criterion for the calculation of the information entropy based on the failure approach index. In addition, combining with the synergetic theory of the principle of maximum information entropy, the evolution equation of the excavation step and information entropy based on the failure approach index of the surrounding rock system in underground mining space are established, and the equations of the general solution and particular solution as well as the expression of the destabilizing excavation step are given. To account for this, the method is applied to analyze the failure states of the floor surrounding rock after the mining of the 71 coal seam in Xutuan Coal Mine and involve the disturbance effect and stability control method of the underlying 72 coal seam roof from the macroscopic and microscopic aspects. Consequently, the validity of the analysis method of synergetic theory of information entropy based on the failure approach index has been verified, which presents an updated approach for the stability evaluation of surrounding rock systems that is of satisfactory capability and value in engineering applications. [ABSTRACT FROM AUTHOR]

A k-connected (resp. k-edge-connected) edge dominating set D of a connected graph G is a subset of E(G) such that G[D] is k-connected (resp. k-edge-connected) and each e ∈ E (G) \ D has at least one neighbor in D. The k-connected edge domination number (resp. k-edge-connected edge domination number) of a graph G is the minimum size of a k-connected (resp. k-edge-connected) edge dominating set of G, and denoted by γ k (G) (resp. γ k ′ (G) ). In this paper, we investigate the relationship between matching number and 2-connected (resp. 2-edge-connected) edge domination number, and prove that for a graph G, if it is 2-edge-connected, then γ 2 ′ (G) ≤ 5 α ′ (G) - 2 , and if it is 2-connected, then γ 2 (G) ≤ 4 α ′ (G) - 1 , where α ′ (G) is the matching number of G. [ABSTRACT FROM AUTHOR]

In general, the ultimate parameter selection method of the failure approach index theory among the three-dimensional problems in geotechnical engineering is unclear in theory, and the symbol convention of the failure approach index in engineering calculation is contrary to the stipulation of the numerical simulation software. Hence, the values of the ultimate plastic shear strain are difficult to determine. To solve this problem, the criterion of positive tension and negative compression and the sequence of the principal stress σ1 ≤ σ2 ≤ σ3 are defined in this paper, and the expression of Mohr–Coulomb yield approach index id deduced. Under the condition of the principal strain sequence ε1 ≤ ε2 ≤ ε3, the formula of the ultimate shear strain is derived using the method of the ultimate strain analysis so as to obtain the simple expression and calculation method of the ultimate plastic shear strain, which has provided the calculation parameters for the three-dimensional ultimate plastic shear strain in the Mohr–Coulomb strain softening model and improved the failure approach index theory. In the light of the aforementioned theory, the ultimate strains of cubic concrete specimens are analyzed, and the obtained ultimate strain values are found consistent with previous research findings, which verifies the correctness and reliability of the ultimate strain analysis method. In addition, it is applied to the quantitative elastic–plastic failure analysis of the section coal pillar in Hengjin coal industry for determining its reasonable retainment width. Consequently, the research results can be embraced as the theoretical basis for the stability analysis of geotechnical materials and exhibits engineering application potential. [ABSTRACT FROM AUTHOR]

The risk evaluation indexes of goaf are multi-source and have complex mutual internal correlations, and there are great differences in the risk identification of goaf from different mines among the various influencing factors. This paper mainly focuses on principal component analysis (PCA) and the differential evolution algorithm (DE), while a multi-classification support vector machine (SVM) is adopted to classify the risks of goaf. Then, the K-fold cross-validation method is used to prevent the overfitting of selection in the model. After the analysis, nine factors affecting the risk identification of goaf in a certain area of East China were determined as the primary influencing factors, and 120 measured goafs were taken as examples for classifying the risks. More specifically, the classification results show that: (1) SVM has the useful ability of generalization, especially when solving the problems of overfitting, and it is easy to fall into the local minima under the conditions of small samples; (2) PCA is employed to realize the intelligent dimensionality reduction and denoising of multi-source impact indicators for goaf risk identification, which immensely improves the prediction accuracy and classification efficiency of the model; (3) after using the DE, the optimal solutions of the problems to be optimized are automatically obtained through the global optimization search mechanism, namely, the kernel function parameter, 'γ', and the penalty factor, 'C', of the SVM, which further verifies that the characteristics of clear logic, strong convergence, and good robustness can be found in the DE. As demonstrated, this method has the advantages of guiding significance and application value for goaf risk identification. [ABSTRACT FROM AUTHOR]

The low intensity and frequent blasting shock wave created by a blasting impact will exert notable destructive effects on an ore rock-filling body. In order to investigate the energy transfer process of the blasting shock wave energy in the filling body and the damage evolution process of the filling body, the software PFC2D was used to numerically simulate and monitor the blasting stress curve of the stope and the energy transfer process. In addition, the incident of the pulse wave with the transmission boundary of the filling body was employed to obtain the fitting curve of the stress wave energy. Simultaneously, the language FISH was adopted to realize the two-dimensional visualization of the damage degree. The results indicate that: (i) the upper side of the filling body is accessible to generate energy superposition, and the shock wave is diffused in a waveform. Moreover, a few large and deep cracks are produced in the central area; (ii) during the blasting process, the damaged area extended continuously from the bottom to the top along the boundary surface of an ore rock-filling body and finally diffused to both sides; (iii) the energy attenuation of the stress wave in an ore body roughly performs the distribution of power function. Consequently, the research results can contribute to the safety evaluation of mining engineering using differential blasting technology. [ABSTRACT FROM AUTHOR]

The k-forcing number of a graph G, denoted by F k (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 G of order n with maximum degree Δ ≥ 2 , Z (G) = F 1 (G) ≤ (Δ - 2) n + 2 Δ - 1 , and this inequality is sharp. Moreover, they posed a conjecture that Z (G) = F 1 (G) = (Δ - 2) n + 2 Δ - 1 if and only if G = C n , G = K Δ + 1 or G = K Δ , 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 if and only if G is connected and F k (G) = n - k if and only if G = K n for k ≤ n - 2 . [ABSTRACT FROM AUTHOR]

Abstract Let S be a set of at least two vertices in a graph G. A subtree T of G is a S-Steiner tree if S ⊆ V (T). Two S -Steiner trees T 1 and T 2 are edge-disjoint (resp. internally disjoint) if E (T 1) ∩ E (T 2) = ∅ (resp. E (T 1) ∩ E (T 2) = ∅ and V (T 1) ∩ V (T 2) = S). Let λ G (S) (resp. κ G (S)) be the maximum number of edge-disjoint (resp. internally disjoint) S -Steiner trees in G , and let λ k (G) (κ k (G)) be the minimum λ G (S) (resp. κ G (S)) for S ranges over all k -subsets of V (G). Clearly, λ 2 (G) (resp. κ 2 (G)) is the classical edge-connectivity λ (G) (resp. connectivity κ (G)). In this paper, we study the λ 3 -connectivity and κ 3 -connectivity of a recursive circulant G , determine λ 3 (G) = δ (G) − 1 for each recursive circulant G , and κ 3 (G) = δ (G) − 1 for each recursive circulant G except G ≅ G (2 m , 2). [ABSTRACT FROM AUTHOR]

Let S be a set of at least two vertices in a graph G . A subtree T of G is a S -Steiner tree if S ⊆ V ( T ) . Two S -Steiner trees T 1 and T 2 are edge-disjoint (resp. internally vertex-disjoint ) if E ( T 1 ) ∩ E ( T 2 ) = ∅ (resp. E ( T 1 ) ∩ E ( T 2 ) = ∅ and V ( T 1 ) ∩ V ( T 2 ) = S ). Let λ G ( S ) (resp. κ G ( S ) ) be the maximum number of edge-disjoint (resp. internally vertex-disjoint) S -Steiner trees in G , and let λ k ( G ) (resp. κ k ( G ) ) be the minimum λ G ( S ) (resp. κ G ( S ) ) for S ranges over all k -subset of V ( G ) . Kriesell conjectured that if λ G ( { x , y } ) ≥ 2 k for any x , y ∈ S , then λ G ( S ) ≥ k . He proved that the conjecture holds for | S | = 3 , 4 . In this paper, we give a short proof of Kriesell’s Conjecture for | S | = 3 , 4 , and also show that λ k ( G ) ≥ 1 k − 1 k ℓ 2 (resp. κ k ( G ) ≥ 1 k − 1 k ℓ 2 ) if λ ( G ) ≥ ℓ (resp. κ ( G ) ≥ ℓ ) in G , where k = 3 , 4 . Moreover, we also study the relation between κ k ( L ( G ) ) and λ k ( G ) , where L ( G ) is the line graph of G . [ABSTRACT FROM AUTHOR]

The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least 2 and , the Steiner distance dG(S) among the vertices of S is the minimum size among all connected subgraphs whose vertex sets contain S. Let n, k be two integers with 2 ≤ k ≤ n. Then the Steiner k-eccentricity ek(v) of a vertex v of G is defined by . Furthermore, the Steiner k-diameter of G is . In 2011, Chartrand, Okamoto and Zhang showed that k − 1 ≤ sdiamk(G) ≤ n − 1. In this paper, graphs with sdiam4( G) = 3, 4, n − 1 are characterized, respectively. [ABSTRACT FROM AUTHOR]

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc( G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that $${\text{trc(G) = 3 if}}\left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) + 1 \leqslant \left| {{\text{E(G)}}} \right| \leqslant \left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) - 1$$ , and $${\text{trc(G)}} \leqslant {\text{6 if }}\left| {{\text{E(G)}}} \right| \geqslant \left( {\begin{array}{*{20}{c}} {n - 2} \\ 2 \end{array}} \right) + 2$$ . Next, we investigate the total rainbow connection numbers of graphs G with | V( G)| = n, diam( G) ≥ 2, and clique number ω( G) = n − s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313-320] is not completely correct, and we provide a complete result for this theorem. [ABSTRACT FROM AUTHOR]

The generalized k -connectivity κ k ( G ) of a graph G , which was introduced by Chartrand et al. (1984), is a generalization of the concept of vertex connectivity. For this generalization, the generalized 2-connectivity κ 2 ( G ) of a graph G is exactly the connectivity κ ( G ) of G . In this paper, let G be a connected graph of order n and let H be a 2-connected graph. For Cartesian product, we show that κ 3 ( G □ H ) ≥ κ 3 ( G ) + 1 if κ ( G ) = κ 3 ( G ) ; κ 3 ( G □ H ) ≥ κ 3 ( G ) + 2 if κ ( G ) > κ 3 ( G ). Moreover, above bounds are sharp. As an example, we show that κ 3 ( C n 1 □ C n 2 □ ⋯ C n k ︷ k ) = 2 k − 1 , where C n i is a cycle. For lexicographic product, we prove that κ 3 ( H ∘ G ) ≥ max { 3 δ ( G ) + 1 , ⌈ 3 n + 1 2 ⌉ } if δ ( G ) < 2 n − 1 3 , and κ 3 ( H ∘ G ) = 2 n if δ ( G ) ≥ 2 n − 1 3 . [ABSTRACT FROM AUTHOR]

In this paper, we consider rainbow connection number of maximal outerplanar graphs (MOPs) on algorithmic aspect. For the (MOP) G , we give sufficient conditions to guarantee that r c ( G ) = d i a m ( G ) . Moreover, we produce the graph with given diameter D and give their rainbow coloring in linear time. X. Deng et al. [4] give a polynomial time algorithm to compute the rainbow connection number of MOPs by the Maximal fan partition method, but only obtain a compact upper bound. J. Lauri [11] proved that, for chordal outerplanar graphs given an edge-coloring, to verify whether it is rainbow connected is NP-complete under the coloring, it is so for MOPs. Therefore we construct Central-cut-spine of MOP G , by which we design an algorithm to give a rainbow edge coloring with at most 2 r a d ( G ) + 2 + c , 0 ≤ c ≤ r a d ( G ) − 2 colors in polynomial time. [ABSTRACT FROM AUTHOR]

The Cayley graphs on the symmetric group plays an important role in the study of Cayley graphs as interconnection networks. Let Cay(Sn,B) be the Cayley graphs generated by transposition-generating trees. It is known that for anyF⊂E(Cay(Sn,B)), if |F|≤n−3 andn≥4, then there exists a hamiltonian cycle in Cay(Sn, B)−F. In this paper, we show that Cay(Sn,B)−Fis bipancyclic if Cay(Sn,B) is not a star graph, forn≥4 and |F|≤n−3. [ABSTRACT FROM PUBLISHER]

The spanning-tree packing number of a graph G is the maximum number of edge-disjoint spanning trees contained in G . In this paper, we obtain a sharp lower bound for the spanning-tree packing number of lexicographic product graphs. [ABSTRACT FROM AUTHOR]