The question of whether the characteristic polynomial of a simple graph is uniquely determined by the characteristic polynomials of its vertex-deleted subgraphs is one of the many unresolved problems in graph reconstruction. In this paper we prove that the characteristic polynomial of a graph is reconstructible from the characteristic polynomials of the vertex-deleted subgraphs of the graph and...

We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power q and constant k, the characteristic polynomial of any loopless, GF (q)representable matroid with tree-width k has no real zero greater than qk−1.

In (West, Discrete Math. 157 (1996) 363-374) it was shown using transfer matrices that the number [Sn(123; 3214)1 of permutations avoiding the pattems 123 and 3214 is the Fibonacci number F2, (as are also IS,(213; 1234)1 and 1S~(213;4123)1 ). We now find the transfer matrix for IS , (123;r , r 1 . . . . . 2,1,r + 1)1, IS,(213;1,2 . . . . . r , r + 1)1, and ISn(213;r + 1,1,2 . . . . . r)l, deter...

In this paper, we are interested in the moments of the characteristic polynomial Zn(x) of the n× n permutation matrices with respect to the uniform measure. We use a combinatorial argument to write down the generating function of E [ ∏p k=1 Z sk n (xk) ] for sk ∈ N. We show with this generating function that limn→∞ E [ ∏p k=1 Z sk n (xk) ] exists for maxk |xk| < 1 and calculate the growth rate ...

Given two ellipsoids, we show that their characteristic equation has at least two negative roots and that the ellipsoids are separated by a plane if and only if their characteristic equation has two distinct positive roots. Furthermore, the ellipsoids touch each other externally if and only if the characteristic equation has a positive double root. An advantage of this characterization is that ...

Abstract. We calculate the fibre integrals of the affine hypersurface in a torus in the form of their Mellin transforms. Especially, our method works efficiently for an affine hypersurface defined by a so called “simpliciable” polynomial. The relations between poles of Mellin transforms of fibre integrals, the mixed Hodge structure of the cohomology of the hypersruface, the hypergeometric diffe...

The problem of robust state feedback design for a linear dynamical system with uncertain (interval) parameters is considered. The designed state feedback controller has to place all the coefficients of the closed loop system characteristic polynomial within assigned closed loop interval characteristic polynomial. A condition is derived using certain known facts about matrix minors and its chara...

In this paper we develop a hybrid symbolic-numeric fast method for solving the secular equation associated with a diagonal plus rank-one eigenvalue problem of order n in O(n) operations and O(n) storage. AMS classification: 65F15

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite st...

The problem of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity was studied recently by Abd-Alla and Ahmed [A.M. Abd-Alla, S.M. Ahmed, Stoneley waves and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity, Appl. Math. Comput. 135 (2003) 187-200], who derived the secular equation of the wave in the implicit f...