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A (d , 1) -total labelling of a simple graph G is an assignment of integers to V (G) ∪ E (G) such that any two adjacent vertices of G receive distinct integers, any two adjacent edges of G receive distinct integers, and a vertex and an edge that are incident in G receive integers that differ by at least d in absolute value. The span of a (d , 1) -total labelling of G is the maximum difference between any two labels. The (d , 1) -total number of G , λ d T (G) , is the minimum span for which G is (d , 1)-total labelled. In this paper, the (d , 1) -total labelling of the Sierpi n ´ ski graph S (n, k), Sierpi n ´ ski gasket graph S n , graphs S + (n , k) and S + + (n , k) are studied, and all of λ d T (S (n , k)) , λ d T (S n) , λ d T (S + (n , k)) and λ d T (S + + (n , k)) for d ≥ k , are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze two new methods for solving the NP-hard absolute value equations (AVE) A x − | x | = b , where A is an arbitrary n × n real matrix and b ∈ R n , in the case, singular value of A exceeds 1. The comparison with other known methods is carried to show the effectiveness of the proposed methods for a variety of randomly generated problems. The ideas and techniques of this paper may stimulate further research. [ABSTRACT FROM AUTHOR]

The energy of a graph is defined as the sum of the absolute values of its eigenvalues. In 1940 Coulson obtained an important integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. Recently several Coulson-type integral formulas have been obtained for various energies and some other invariants of graphs based on eigenvalues. For a complex polynomial ϕ ( z ) = ∑ k = 0 n a k z n − k = a 0 ∏ k = 1 n ( z − z k ) of degree n and a real number α , the general energy of ϕ ( z ), denoted by E α ( ϕ ), is defined as ∑ z k ≠ 0 | z k | α when there exists k 0 ∈ { 1 , 2 , … , n } such that z k 0 ≠ 0 , and 0 when z 1 = ⋯ = z n = 0 . In this paper we give Coulson-type integral formulas for the general energy of polynomials whose roots are all real numbers in the case that α ∈ Q . As a consequence of this result, we obtain an integral formula for the 2 l -th spectral moment of a graph. Furthermore, we show that our formulas hold when α is an irrational number with 0 < | α | < 2 and do not hold with | α | > 2. [ABSTRACT FROM AUTHOR]

• In this paper, the nonlinear fault detection observer is constructed and the observer gain function is derivated by choosing the novel Lyapunov functions. The predetermined estimation accuracy-dependent nonnegative function is introducing in the last step of recursive algorithm. • The threshold design problem is solved by suppressing the residual within the predetermined output estimation accuracy range during the implementation process of observer. • The tight thresholds ϵ and ϵ are predetermined under strategic designing scheme. It is straightforward to implement by comparison with the existing empirical or calculation methods of threshold. This paper presents an fault detection (FD) method for a class of uncertain nonlinear systems with unmatched nonlinear fault functions and disturbances. A recursive FD observer is designed with predetermined and small output estimation error. The nonlinear observer gain function is achieved by introducing predetermined output estimation accuracy-dependent nonnegative functions. Combining Lyapunov functions, it is shown that the absolute value of the residual signal is equal or lesser than tight threshold before fault occurrence. The FD scheme is proposed following fault detectability analysis, simulation example indicates the validity of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose an SOR-like iteration method for solving the absolute value equation (AVE), which is obtained by reformulating equivalently the AVE as a two-by-two block nonlinear equation. The convergence results of the proposed iteration method are proved under certain assumptions imposed on the involved parameter. Numerical experiments are given to demonstrate the feasibility, robustness and effectiveness of the SOR-like iteration method. [ABSTRACT FROM AUTHOR]

In 1978, Gutman proposed the concept of graph energy, defined as the sum of the absolute values of eigenvalues of the adjacency matrix of a molecular graph, which is related to the energy of π -electrons in conjugated hydrocarbons. Recently, Gutman and Wagner proposed the concept of matching energy and pointed out that the chemical applications of matching energy go back to the 1970s. In this paper, we study the extremal values of the matching energy and characterize the graphs with minimal matching energy among all tricyclic graphs with a given diameter. Our methods can help to find more extremal values for other classes of molecular networks and the results suggest the structures with extremal energies. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } . The Randić matrix of G , denoted by R ( G ), is defined as the n × n matrix whose ( i, j )-entry is ( d i d j ) − 1 2 if v i and v j are adjacent and 0 for another cases. Let the eigenvalues of the Randić matrix R ( G ) be ρ 1 ≥ ρ 2 ≥ ⋅⋅⋅ ≥ ρ n which are the roots of the Randić characteristic polynomial ∏ i = 1 n ( ρ − ρ i ) . The Randić energy RE of G is the sum of absolute values of the eigenvalues of R ( G ). In this paper, we compute the Randić characteristic polynomial and the Randić energy for specific graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of absolute value equations associated with circular cone (CCAVE for short), which is a generalization of the absolute value equations discussed recently in the literature, analogously to the fact that circular cone is the very generalization of the second-order cone. We show that the CCAVE is equivalent to a class of circular cone linear complementarity problems, and hence generalize the well-known equivalence between absolute value equations and linear complementarity problems. Useful properties of the generalized differential of the absolute value function over the circular cone are investigated, which helps to propose a generalized Newton method for solving the CCAVE. The convergence of this method is established under mild conditions, as well as the efficiency of which is illustrated by some preliminary numerical results. [ABSTRACT FROM AUTHOR]

The energy of a simple graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix. Jianping found the structure of trees which have maximal energy among all trees of diameter 4 whose centers are of degree t . This paper generalizes the result. We determine the structure of the general tree of diameter 4 and maximal energy. [ABSTRACT FROM AUTHOR]

Finding the solution of the absolute value equation (AVE) A x − | x | = b has attracted much attention in recent years. In this paper, we propose a relaxed nonlinear PHSS-like iterative method, which is more efficient than the Picard-HSS iterative method for the AVE, and is a generalization of the nonlinear HSS-like iteration method for the AVE. By using the theory of nonsmooth analysis, we prove the convergence of the relaxed nonlinear PHSS-like iterative method for the AVE. Numerical experiments are given to demonstrate the feasibility, robustness and effectiveness of the relaxed nonlinear HSS-like method. [ABSTRACT FROM AUTHOR]

• A novel distance-based topological index namely the Mostar index ofgraphsis studied. • Some sharp bounds on this index of trees with given degree sequence are established. • A regression analysis of the linear models of acentric factors and entropy of octaneisomers and boiling points of benzenoid hydrocarbons with the distance based indiceshave been carried out and also compared the linear models. • The linear model obtained, based on the Mostar index, is better than the similar modelsbased on the other distance-based indices. For a given graph G , the Mostar index Mo (G) is the sum of absolute values of the differences between n u (e) and n v (e) over all edges e = u v of G , where n u (e) and n v (e) are, respectively, the number of vertices of G lying closer to u than to v and the number of vertices of G lying closer to v than to u. The degree sequence of a tree is the sequence of the degrees (in descending order) of the non-leaf vertices. This paper determines those trees with a given degree sequence which have the greatest Mostar index. Consequently, all extremal trees with the greatest Mostar index are obtained in the sets of all trees of order n with the maximum degree, the number of leaves, the independence number and the matching number, respectively. On the other hand, some properties of trees with a given degree sequence which have the least Mostar index are given. We also determine those trees with exactly n − 3 or n − 4 leaves when the degree sequence is given, which have the least Mostar index. At last some numerical results are discussed, in which we calculate the Mostar indices of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons; We compare their Mostar indices with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the Mostar index is better than or as good as the models corresponding to the other distance-based indices. [ABSTRACT FROM AUTHOR]