For a graph, the first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Denote by Gn,k the set of graphs with n vertices and k cut edges. In this paper, we showed the types of graphs with the largest and the second largest M1 and M2 among Gn,k .

Let G be a simple graph with order n and size m. The quantity $$M_1(G)=\sum _{i=1}^{n}d^2_{v_i}$$ is called the first Zagreb index of G, where $$d_{v_i}$$ degree vertex $$v_i$$ , for all $$i=1,2,\dots ,n$$ . signless Laplacian matrix $$Q(G)=D(G)+A(G)$$ A(G) D(G) denote, respectively, adjacency diagonal degrees G. $$q_1\ge q_2\ge \dots \ge q_n\ge 0$$ eigenvalues largest eigenvalue $$q_1$$ spectr...

Let K be a totally real Galois number field. C. J. Hillar proved that if f ∈ Q[x1, . . . , xn] is a sum of m squares in K[x1, . . . , xn], then f is a sum of N(m) squares in Q[x1, . . . , xn], where N(m) ≤ 2[K:Q]+1 · `[K:Q]+1 2 ́ ·4m, the proof being constructive. We show in fact that N(m) ≤ (4[K : Q]−3)·m, the proof being constructive as well.

We present a new approach to evaluating combinatorial sums by using finite differences. Let {ak}k=0 and {bk}k=0 be sequences with the property that ∆bk = ak for k ≥ 0. Let gn = ∑n k=0 ( n k ) ak, and let hn = ∑n k=0 ( n k ) bk. We derive expressions for gn in terms of hn and for hn in terms of gn. We then extend our approach to handle binomial sums of the form ∑n k=0 ( n k ) (−1)ak, ∑ k ( n 2k ...

The classical Brunn-Minkowski theory for convex bodies was developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. As a special case of mixed volumes, the Quermassintegrals are important geometrical quantities of a convex body, and surface area measures are local versions of Quermassintegrals. The Christoffel-Minkowski problem concerns with the existen...

In this article, we study the vertices D∗D∗π and D∗D∗ sK in the framework of the light-cone QCD sum rules approach. The strong coupling constants gD∗D∗π, gD∗D∗ sK play an important role in understanding the final-state rescattering effects in the hadronic B decays. And they are related to the basic parameter g in the heavy quark effective Lagrangian, the existing estimations for g vary in a lar...

Given a set of point P in Rd, a clustering problem is to partition P into k subsets {P1, P2, · · · , Pk} in such a way that a given objective function is minimized. The most studied cost functions for a cluster, μ(Pi), are maximum or average radius of Pi, maximum diameter of Pi, and maximum width of Pi. The overall objective function is ⊕ μ(Pi), where ⊕ is typically the Lp-norm operator. The mo...

Let A = {a1, a2, . . . } (a1 < a2 < . . . ) be a xed in nite sequence of positive integers, and let Rk(n) denote the number of solutions of ai1 + ai2 + · · · + aik = n, ai1 ∈ A, . . . , aik ∈ A. For k = 2, P. Erd®s and A. Sárközy proved if F (n) is a nice arithmetic function then there exists a sequence A such that |R2(n)−F (n)| (F (n) log n)1/2. The aim of this paper is to extend their result ...