Search

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph G , are denoted by D (G) , D Q (G) = D (G) + Tr (G) and D L (G) = Tr (G) − D (G) , respectively, where Tr (G) is the diagonal matrix of vertex transmission. Ger s ̌ gorin discs for any n × n square matrix A = [ a i j ] are the discs { z ∈ ℂ : | z − a i i | ≤ R i (A) } , where R i (A) = ∑ j ≠ i | a i j | , i = 1 , 2 , ... , n. The famous Ger s ̌ gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger s ̌ gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger s ̌ gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger s ̌ gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger s ̌ gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2]. [ABSTRACT FROM AUTHOR]

For a simple connected graph G , let D (G) and T (G) be the distance matrix and the diagonal matrix of the vertex transmissions, respectively. The convex linear combination D α (G) of D (G) and T (G) is defined as, D α (G) = α T (G) + (1 − α) D (G) , 0 ≤ α ≤ 1. The matrix D α (G) , known as generalized distance matrix, is effective in merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral radius and energy of the generalized distance matrix D α (G) of a graph G. We obtain bounds for the generalized distance spectral radius and generalized distance energy of connected graphs in terms of various parameters associated with the structure of graph. [ABSTRACT FROM AUTHOR]

For a simple connected graph G , the generalized distance matrix D α (G) is defined as D α (G) = α Tr (G) + (1 − α) D (G) , 0 ≤ α ≤ 1 , where Tr (G) and D (G) are, respectively, the diagonal matrix of vertex transmission degrees and the distance matrix of G. In this paper, we will study the spectral radius of the generalized distance matrix D α (G) of a connected graph G. We obtain some upper and lower bounds for the generalized distance spectral radius in terms of various parameters, like the vertex transmission degrees, the average transmission 2-degrees, etc., associated with the structure of the graph. We characterize the extremal graphs attaining these bounds. Further, we show that our bounds improve some previously known bounds existing in the literature. [ABSTRACT FROM AUTHOR]

Given a connected graph G , the distance Laplacian matrix D L (G) is defined as D L (G) = Tr (G) − D (G) , and the distance signless Laplacian matrix D Q (G) is defined as D Q (G) = Tr (G) + D (G) , where Tr (G) is the transmission matrix of G and D (G) is the distance matrix of G. The Indu–Bala product of two graphs G 1 and G 2 , denoted by G 1 ▾ G 2 , was introduced in (G. Indulal and R. Balakrishnan, Distance spectrum of Indu–Bala product of graphs,  AKCE Int. J. Graph Comb. 13(3) (2016) 230–234). In this paper, we first obtain the distance Laplacian spectrum of G 1 ▾ G 2 in terms of Laplacian spectra of G 1 and G 2 . We then obtain the distance signless Laplacian spectrum of G 1 ▾ G 2 in terms of signless Laplacian spectra of G 1 and G 2 . We construct pair of graphs which are distance Laplacian co-spectral as well as pair of graphs which are distance signless Laplacian co-spectral. We further find the metric dimension of G 1 ▾ G 2 in terms of metric dimensions of G 1 and G 2 . Finally, we provide a problem for future research. [ABSTRACT FROM AUTHOR]

For a connected graph G of order n with normalized signless Laplacian eigenvalues γ 1 + ≥ γ 2 + ≥ ⋯ ≥ γ n + ≥ 0 , the normalized signless Laplacian resolvent energy of G is defined as ERNS (G) = ∑ i = 1 n 1 3 − γ i + . In this paper, we derive new inequalities on ERNS (G) as well as relations on ERNS (G) with Randić (normalized) incidence energy. We also deduced that some of our results improve the existing results in the literature. [ABSTRACT FROM AUTHOR]

This research uses the Sinc Collocation method to numerically examine the Degasperis–Procesi and Benjamin–Bona–Mahony equations, achieving a high level of precision and accuracy on computational grounds with a variety of mesh points. The proposed technique involves global collocation using Sinc bases function (SBF) as an activation function. Initially, the time derivative discretization has been performed through finite difference scheme and the remaining spatial derivatives are approximated by θ -weighted scheme to minimize the complexity in numerical computations. The main contribution of research is in modifying the proposed scheme to overcome the complexity involved in the whole process of finding solutions with a cardinal expansion of Sinc functions in nonlinear and singular systems. The solutions of Degasperis–Procesi and Benjamin–Bona–Mahony stiff nonlinear systems are illustrated numerically and graphically and the validation and verification of the obtained results are examined through stability analysis by calculating the spectral radius by bounding the value of θ. A comparative study is presented with some of the well-known numerical techniques, which assured the accuracy and reliability of our technique. [ABSTRACT FROM AUTHOR]

The existing rate of convergence of series requires a known limit of that series. Direct computation of Brun's constant as the sum of twin prime reciprocals is challenging as the convergence is agonizingly slow. Here, I introduce the relative increase of the partial sum of a series as a new rate of convergence regardless of the limit of the series. With this concept, I extract a slowly convergent part from the original sum for Brun's constant so that the remaining one converges faster. Computed data demonstrate the usefulness of the concept. As a result, I obtain an improved lower bound for Brun's constant using less computing resource and establish an approximate mapping that gives a new estimate for the constant at 1.89558413. [ABSTRACT FROM AUTHOR]

We prove a sharp higher differentiability result for local minimizers of functionals of the form F(w, Ω) = ∫Ω [F(x,Dw(x)) - f(x) · w(x)]dx with non-autonomous integrand F(x, ξ) which is convex with respect to the gradient variable, under p-growth conditions, with 1 < p < 2. The main novelty here is that the results are obtained assuming that the partial map x ... DξF(x, ξ) has weak derivatives in some Lebesgue space Lq and the datum f is assumed to belong to a suitable Lebesgue space Lr. We also prove that it is possible to weaken the assumption on the datum f and on the map x ... DξF(x, ξ), if the minimizers are assumed to be a priori bounded. [ABSTRACT FROM AUTHOR]

Let Ta be a pseudo-differential operator defined by the symbol a ... with 0 < ... ≤ 1, 0 ≤ δ < 1. It is shown that if ..., then the operator satisfies the following pointwise estimate: ... for all x ∈ ℝn and all Schwartz function u. Here, u... denotes the John-Nirenberg sharp function of u and M stands for the Hardy--Littlewood maximal operator. [ABSTRACT FROM AUTHOR]

We define the path Laplacian matrix and the path signless Laplacian matrix of a simple connected graph G as P ℒ (G) = T r P (G) − P (G) and P (G) = T r P (G) + P (G) , respectively, where P (G) is the path matrix and T r P (G) is the diagonal matrix of the vertex transmissions. The generalized path matrix is P α (G) = α T r P (G) + (1 − α) P (G) , for 0 ≤ α ≤ 1 and ρ 1 α ≥ ρ 2 α ≥ ⋯ ≥ ρ n α are the eigenvalues of P α (G). The generalized path energy can be expressed as E P α (G) = ∑ i = 1 n ρ i α − 2 α P W (G) n , where P W (G) is the path Wiener index of G. We give basic properties of generalized path matrix P α (G). Also, some upper and lower bounds of the generalized path energy of some graphs are studied. [ABSTRACT FROM AUTHOR]