The spectra of some trees and bounds for the largest eigenvalue of any tree

Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n_k−j+1 and d_k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (n_k = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d₁, d₂, …, d_k−1, d_k ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.     

Fonction asymptotique de Samuel des sections hyperplanes et multiplicité

Let (A,m_A,k) be a local noetherian ring and I an m_A-primary ideal. The asymptotic Samuel function (with respect to I) : A?R∪{+∞} is defined by , ∀x∈A. Similarly, one defines, for another ideal J, as the minimum of as x varies in J. Of special interest is the rational number . We study the behavior of the asymptotic Samuel function (with respect to I) when passing to hyperplane sections of A as one does for the theory of mixed multiplicities.     

On the existence of graphs of diameter two and defect two

In the context of the degree/diameter problem, the ‘defect’ of a graph represents the difference between the corresponding Moore bound and its order. Thus, a graph with maximum degree d and diameter two has defect two if its order is n=d²−1. Only four extremal graphs of this type, referred to as (d,2,2)-graphs, are known at present: two of degree d=3 and one of degree d=4 and 5, respectively. In this paper we prove, by using algebraic and spectral techniques, that for all values of the degree d within a certain range, (d,2,2)-graphs do not exist.The enumeration of (d,2,2)-graphs is equivalent to the search of binary symmetric matrices A fulfilling that AJ_n=dJ_n and A²+A+(1−d)I_n=J_n+B, where J_n denotes the all-one matrix and B is the adjacency matrix of a union of graph cycles. In order to get the factorization of the characteristic polynomial of A in Q[x], we consider the polynomials F_i,d(x)=f_i(x²+x+1−d), where f_i(x) denotes the minimal polynomial of the Gauss period , being ζ_i a primitive ith root of unity. We formulate a conjecture on the irreducibility of F_i,d(x) in Q[x] and we show that its proof would imply the nonexistence of (d,2,2)-graphs for any degree d>5.     

Laplacian energy of a graph

Let G be a graph with n vertices and m edges. Let λ₁, λ₂, … , λ_n be the eigenvalues of the adjacency matrix of G, and let μ₁, μ₂, … , μ_n be the eigenvalues of the Laplacian matrix of G. An earlier much studied quantity is the energy of the graph G. We now define and investigate the Laplacian energy as . There is a great deal of analogy between the properties of E(G) and LE(G), but also some significant differences.     

The minimum distance of sets of points and the minimum socle degree

Let K be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a hyperplane. Let be the maximum number of points of Γ contained in any hyperplane, and let . If I⊂R=K[x₀,…,x_n] is the ideal of Γ, then in Tohaˇneanu (2009) [12] it is shown that for n=2,3, d(Γ) has a lower bound expressed in terms of some shift in the graded minimal free resolution of R/I. In these notes we show that this behavior holds true in general, for any n≥2: d(Γ)≥A_n, where A_n=min{a_i−n} and ⊕_iR(−a_i) is the last module in the graded minimal free resolution of R/I. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (2010) [10].     

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n_±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A^?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.     

Approximating classes of sequences: The Hermitian case

Given a sequence {A_n} of matrices A_n of increasing dimension d_n with d_k>d_q for k>q, k,q∈N, we recently introduced the concept of approximating class of sequences (a.c.s.) in order to define a basic approximation theory for matrix sequences. We have shown that such a notion is stable under inversion, linear combinations, and product, whenever natural and mild conditions are satisfied. In this note we focus our attention on the Hermitian case and we show that is an a.c.s. for {f(A_n)}, if is an a.c.s. for {A_n}, {A_n} is sparsely unbounded, and f is a suitable continuous function defined on R. We also discuss the potential impact and future developments of such a result.     

On the saturation sequence of the rational normal curve

Let denote the rational normal curve of order d. Its homogeneous defining ideal admits an SL₂-stable filtration J₂⊆J₄⊆…⊆I_C by sub-ideals such that the saturation of each J_2q equals I_C. Hence, one can associate to d a sequence of integers (α₁,α₂,…) which encodes the degrees in which the successive inclusions in this filtration become trivial. In this paper we establish several lower and upper bounds on the α_q, using inter alia the methods of classical invariant theory.     

Castelnuovo-Mumford regularity for complexes and weakly Koszul modules

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and the category of finitely generated graded left A-modules. Following Jørgensen, we define the Castelnuovo-Mumford regularity of a complex in terms of the local cohomologies or the minimal projective resolution of M^•. Let A^! be the quadratic dual ring of A. For the Koszul duality functor , we have . Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A^!. As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E=?〈y₁,…,y_d〉, d≥3, then the (d−2)nd syzygy of E/J is weakly Koszul.     

Cayley digraphs with normal adjacency matrices

In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset S. It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group G is normal iff G is either abelian or has the form for some non-negative integer n, where Q₈ is the quaternion group and is the abelian group of order 2ⁿ and exponent 2.     

Eigenvalues and forbidden subgraphs I

Suppose a graph G have n vertices, m edges, and t triangles. Letting λ_n(G) be the largest eigenvalue of the Laplacian of G and μ_n(G) be the smallest eigenvalue of its adjacency matrix, we prove that

     

Stability of associated primes of integral closures of monomial ideals

Let I be a monomial ideal of a polynomial ring R=K[X₁,…,Xr] and d(I) the maximal degree of minimal generators of I. In this paper, we explicitly determine a number n₀ in terms of r and d(I) such that for all n?n₀. Furthermore, our n₀ is almost sharp.     

The Laplacian spectral radius of some bipartite graphs

In this paper, we study the largest Laplacian spectral radius of the bipartite graphs with n vertices and k cut edges and the bicyclic bipartite graphs, respectively. Identifying the center of a star K_1,k and one vertex of degree n of K_m,n, we denote by the resulting graph. We show that the graph (1?k?n-4) is the unique graph with the largest Laplacian spectral radius among the bipartite graphs with n vertices and k cut edges, and (n?7) is the unique graph with the largest Laplacian spectral radius among all the bicyclic bipartite graphs.     

Configurations in abelian categories. III. Stability conditions and identities

This is the third in a series on configurations in an abelian category A. Given a finite poset (I,?), an (I,?)-configuration(σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of configurations in A, using the theory of Artin stacks. It showed well-behaved moduli stacks ObjA,MA(I,?) of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA.This paper introduces (weak) stability conditions(τ,T,?) on A. We show the moduli spaces , , of τ-semistable, indecomposable τ-semistable and τ-stable objects in class α are constructible sets in ObjA, and some associated configuration moduli spaces constructible in MA(I,?), so their characteristic functions and are constructible.We prove many identities relating these constructible functions, and their stack function analogues, under pushforwards. We introduce interesting algebras of constructible and stack functions, and study their structure. In the fourth paper we show are independent of (τ,T,?), and construct invariants of A,(τ,T,?).     

Configurations in abelian categories. IV. Invariants and changing stability conditions

This is the last in a series on configurations in an abelian category A. Given a finite poset (I,?), an (I,?)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of configurations in A, using Artin stacks. It showed well-behaved moduli stacks ObjA,MA(I,?) of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA.The third introduced stability conditions(τ,T,?) on A, and showed the moduli space of τ-semistable objects in class α is a constructible subset in ObjA, so its characteristic function is a constructible function. It formed algebras , , , of constructible and stack functions on ObjA, and proved many identities in them.In this paper, if (τ,T,?) and are stability conditions on A we write in terms of the , and deduce the algebras are independent of (τ,T,?). We study invariants or Iss(I,?,κ,τ) ‘counting’ τ-semistable objects or configurations in A, which satisfy additive and multiplicative identities. We compute them completely when A=mod-KQ or A=coh(P) for P a smooth curve. We also find invariants with special properties when A=coh(P) for P a smooth surface with nef, or a Calabi-Yau 3-fold.     

Balanced Leonard pairs

Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A^∗ : V → V that satisfy the following two conditions:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

Let (respectively v₀, v₁, … , v_d) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ? i ? d, let a_i denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of v_i, when we write A^∗v_i as a linear combination of v₀, v₁, … , v_d.In this paper we show a₀ = a_d if and only if . Moreover we show that for d ? 1 the following are equivalent; (i) a₀ = a_d and a₁ = a_d−1; (ii) and ; (iii) a_i = a_d−i and for 0 ? i ? d. These give a proof of a conjecture by the second author. We say A, A^∗ is balanced whenever a_i = a_d−i and for 0 ? i ? d. We say A,A^∗ is essentially bipartite (respectively essentially dual bipartite) whenever a_i (respectively ) is independent of i for 0 ? i ? d. Observe that if A, A^∗ is essentially bipartite or dual bipartite, then A, A^∗ is balanced. For d ≠ 2, we show that if A, A^∗ is balanced then A, A^∗ is essentially bipartite or dual bipartite.