Hamilton Properties of Domination Critical Graphs

Ewa·Wojcicka [1] has proved that the connected 3 γ critical graphs has a H path and has put forward to such a conjecture: Connected 3 γ critical graphs without endpoints are H graphs. In this paper,we prove that if G is a connected 3 γ critical graph without endpoints and has a H paht ap →a such that d(a,b)=3, then G is a H graph. The result partially solves Ewa. Wojcickas conjecture.

(a, b, k )-critical graphs

THE graphs considered in this letter will be finite and undirected simple graphs. Let G be a graph with vertex set V(G) and edge set E(G) . The minimum degree of G is denoted by δ(G) . Notation and definition not given in this letter can be found in reference [1]. For S V(G) the induced subgraph of G by S is denoted by G[S] . For any vertex

OnFractional(g,f,n′,m)-Critical Covered Graphs

The main contribution in this article is threefold:(1)we show the necessary and sufficient condition for graphs to be fractional(g,f)-covered which can be expressed in different forms,and extended to fractional(g,f,m)-covered graphs;(2)the concept of fractional-critical covered graph is put forward and its necessary and sufficient condition is given;(3)we present the degree condition for a graph to be fractional(g,f,n′,m)-critical covered,and show that degree bound is sharp when m is small.Moreover,the related result in fractional(a,b,n′,m)-critical covered setting is also verified.

Graphs Whose Critical Groups Have Larger Rank

The critical group C（G） of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r（G） be the minimum number of generators （i.e., the rank） of the group C（G） and β（G） be the number of independent cycles of G. In this paper, some forbidden induced subgraphs axe given for r（G） = n - 3 and all graphs with r（G） = j3（G） = n - 3 are characterized

Star-critical Ramsey Numbers of Wheels Versus Odd Cycles

Let K_(1,k)be a star of order k+1 and K_(n)■K_(1,k)the graph obtained from a complete graph K_(n)and an additional vertex v by joining v to k vertices of K_(n).For graphs G and H,the star-critical Ramsey number r_(*)(G,H)is the minimum integer k such that any red/blue edge-coloring of K_(r-1)■K_(1,k)contains a red copy of G or a blue copy of H,where r is the classical Ramsey number R(G,H).Let C_(m)denote a cycle of order m and W_(n)a wheel of order n+1.Hook(2010)proved that r_(*)(W_(n),C_3)=n+3 for n≥6.In this paper,we show that r_(*)(W_(n),C_(m))=n+3 for m odd,m≥5 and n≥3(m-1)/2+2.

Cheeger's cut, maxcut and the spectral theory of1-Laplacian on graphs

This is primarily an expository paper surveying up-to-date known results on the spectral theory of1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.

Hybrid stacking ensemble algorithm and simulated annealing optimization for stability evaluation of underground entry-type excavations

The stability of underground entry-type excavations(UETEs)is of paramount importance for ensuring the safety of mining operations.As more engineering cases are accumulated,machine learning(ML)has demonstrated great potential for the stability evaluation of UETEs.In this study,a hybrid stacking ensemble method aggregating support vector machine(SVM),k-nearest neighbor(KNN),decision tree(DT),random forest(RF),multilayer perceptron neural network(MLPNN)and extreme gradient boosting(XGBoost)algorithms was proposed to assess the stability of UETEs.Firstly,a total of 399 historical cases with two indicators were collected from seven mines.Subsequently,to pursue better evaluation performance,the hyperparameters of base learners(SVM,KNN,DT,RF,MLPNN and XGBoost)and meta learner(MLPNN)were tuned by combining a five-fold cross validation(CV)and simulated annealing(SA)approach.Based on the optimal hyperparameters configuration,the stacking ensemble models were constructed using the training set(75%of the data).Finally,the performance of the proposed approach was evaluated by two global metrics(accuracy and Cohen’s Kappa)and three within-class metrics(macro average of the precision,recall and F1-score)on the test set(25%of the data).In addition,the evaluation results were compared with six base learners optimized by SA.The hybrid stacking ensemble algorithm achieved better comprehensive performance with the accuracy,Kappa coefficient,macro average of the precision,recall and F1-score were 0.92,0.851,0.885,0.88 and 0.883,respectively.The rock mass rating(RMR)had the most important influence on evaluation results.Moreover,the critical span graph(CSG)was updated based on the proposed model,representing a significant improvement compared with the previous studies.This study can provide valuable guidance for stability analysis and risk management of UETEs.However,it is necessary to consider more indicators and collect more extensive and balanced dataset to validate the model in future.

Stability prediction of underground entry-type excavations based on particle swarm optimization and gradient boosting decision tree

The stability of underground entry-type excavations will directly affect the working environment and the safety of staff.Empirical critical span graphs and traditional statistics learning methods can not meet the requirements of high accuracy for stability assessment of entry-type excavations.Therefore,this study proposes a new prediction method based on machine learning to scientifically adjust the critical span graph.Accordingly,the particle swarm optimization(PSO)algorithm is used to optimize the core parameters of the gradient boosting decision tree(GBDT),abbreviated as PSO-GBDT.Moreover,the classification performance of eight other classifiers including GDBT,k-nearest neighbors(KNN),two kinds of support vector machines(SVM),Gaussian naive Bayes(GNB),logistic regression(LR)and linear discriminant analysis(LDA)are also applied to compare with the proposed model.Findings revealed that compared with the other eight models,the prediction performance of PSO-GBDT is undoubtedly the most reliable,and its classification accuracy is up to 0.93.Therefore,this model has great potential to provide a more scientific and accurate choice for the stability prediction of underground excavations.In addition,each classification model is used to predict the stability category of several grid points divided by the critical span graph,and the updated critical span graph of each model is discussed in combination with previous studies.The results show that the PSO-GBDT model has the advantages of being scientific,accurate and efficient in updating the critical span graph,and its output decision boundary has strict theoretical support,which can help mine operators make favorable economic decisions.