Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

Abelian group matrices over the p-adic and rational integers

Let G be a finite abelian group of order n. Let Z and Q denote the rational integers and rationals, respectively. A group matrix for G over Z (or Q) is an n-square matrix of the form Σ_{g ∈ G}a_gP(g), where a_g ∈ Z (or Q) and P is the regular representation of G so that P(g) is an n-square permutation matrix and P(gh) = P(g)P(h) for all g, h ∈ G. It is known that if M is an arbitrary positive definite unimodular matrix over Z then there exists a matrix A over Q such that M = A^τA, where τ denotes transposition. This paper proves that the exact analogue of this theorem holds if one demands that M and A be group matrices for G over Z and Q, respectively. Furthermore, if M is a group matrix for G over the p-adic integers then necessary and sufficient conditions are given for the existence of a group matrix A for G over the p-adic numbers such that M = A^τA.     

Main Q-eigenvalues and generalized Q-cospectrality of graphs

Let Q ^G denote the signless Laplacian matrix of a graph G. An eigenvalue μ of Q ^G is said to be a main Q-eigenvalue of G if μ has an eigenvector which is not orthogonal to an all-ones vector e. We give some basic properties of main Q-eigenvalues. For a graph G of order n, G is called Q-controllable if G has n distinct main Q-eigenvalues. We show that a graph H is generalized Q-cospectral with a Q-controllable G if and only if H is Q-controllable and there exists a unique rational orthogonal matrix R such that R _e = e, Q _H = R ^? Q _G R.     

Bounds on the index of the signless Laplacian of a graph

Let G=(V,E) be a simple, undirected graph of order n and size m with vertex set V, edge set E, adjacency matrix A and vertex degrees Δ=d₁≥d₂≥?≥d_n=δ. The average degree of the neighbor of vertex v_i is . Let D be the diagonal matrix of degrees of G. Then L(G)=D(G)−A(G) is the Laplacian matrix of G and Q(G)=D(G)+A(G) the signless Laplacian matrix of G. Let μ₁(G) denote the index of L(G) and q₁(G) the index of Q(G). We survey upper bounds on μ₁(G) and q₁(G) given in terms of the d_i and m_i, as well as the numbers of common neighbors of pairs of vertices. It is well known that μ₁(G)≤q₁(G). We show that many but not all upper bounds on μ₁(G) are still valid for q₁(G).     

On conjectures involving second largest signless Laplacian eigenvalue of graphs

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G)=D(G)-A(G) and the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetkovi? et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.     

Morphing polyhedra with parallel faces: Counterexamples

Two simple polyhedra P and Q (not necessarily convex) are parallel if they share the same edge graph G and each face of P has the same outward-facing unit normal as the corresponding face in Q. Parallel polyhedra P and Q admit a parallel morph if the vertices can be moved in a continuous manner taking us from P to Q such that at all times the intermediate polyhedron determined by the vertex configuration and graph G is both simple and parallel with P (and Q). In this note, we show that even for very restrictive classes of orthogonal polyhedra, there exist parallel polyhedra that do not admit a parallel morph.     

Variations on a Theme of Cline and Donkin

Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q ^g of Q by g, for every g?∈?G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G _r (r?≥?1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ^?⊕?n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules.     

Sharp bounds for the largest eigenvalue of the signless Laplacian of a graph

Let G be a simple connected graph with n vertices and m edges. Denote the degree of vertex v_i by d(v_i). The matrix Q(G)=D(G)+A(G) is called the signless Laplacian of G, where D(G)=diag(d(v₁),d(v₂),…,d(v_n)) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let q₁(G) be the largest eigenvalue of Q(G). In this paper, we first present two sharp upper bounds for q₁(G) involving the maximum degree and the minimum degree of the vertices of G and give a new proving method on another sharp upper bound for q₁(G). Then we present three sharp lower bounds for q₁(G) involving the maximum degree and the minimum degree of the vertices of G. Moreover, we determine all extremal graphs which attain these sharp bounds.     

The depression of a graph and the diameter of its line graph

An edge ordering of a graph G=(V,E) is an injection f:E→Q⁺ where Q⁺ is the set of positive rational numbers. A (simple) path λ for which f increases along its edge sequence is an f-ascent, and a maximal f-ascent if it is not contained in a longer f-ascent. The depression ε(G) of G is the least integer k such that every edge ordering of G has a maximal ascent of length at most k.It has been shown in [E.J. Cockayne, G. Geldenhuys, P.J.P. Grobler, C.M. Mynhardt, J. van Vuuren, The depression of a graph, Utilitas Math. 69 (2006) 143-160] that the difference may be made arbitrarily large. We prove that the difference can also be arbitrarily large, thus answering a question raised in [E.J. Cockayne, G. Geldenhuys, P.J.P. Grobler, C.M. Mynhardt, J. van Vuuren, The depression of a graph, Utilitas Math. 69 (2006) 143-160].     

Asymptotics of functions satisfying the γ-weak inequality with applications to uniformly bounded representations

Let G be a real reductive group of class $\text{H}$, and π a uniformly bounded representation of G on a Hilbert space having infinitesimal character. We then show that the K-finite matrix elements of π decay “exponentially” on G provided that the infinitesimal character of π is in general position. Further we show that π is infinitesimally equivalent to a subquotient of a cuspidal principal series representation π_Q,ω,ν where ν belongs to a tube domain defined by ?Q. These facts follow from the asymptotics of functions satisfying the γ-weak inequality.     

Edge-bipancyclicity of a hypercube with faulty vertices and edges

A bipartite graph G=(V,E) is said to be bipancyclic if it contains a cycle of every even length from 4 to |V|. Furthermore, a bipancyclic G is said to be edge-bipancyclic if every edge of G lies on a cycle of every even length. Let F_v (respectively, F_e) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q_n. In this paper, we show that every edge of Q_n-F_v-F_e lies on a cycle of every even length from 4 to 2ⁿ-2|F_v| even if |F_v|+|F_e|?n-2, where n?3. Since Q_n is bipartite of equal-size partite sets and is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free cycle obtained are worst-case optimal.     

Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovi? et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q₂) and the index (λ₁) of graph G (see also Aouchiche and Hansen [1]):

     

Some new characterizations of solvable PST-groups

A subgroup H of a finite group G is said to be ??-semipermutable in G if it permutes with all the Sylow subgroups Q of G such that (|H|, |Q|) = 1 and (|H|, |Q ^G |) ?? 1. A rather remarkable result of Lukyanenko and Skiba (Rend Semin Mat Univ Padova, 124:231?C246, 2010) is: a finite solvable group G is a PST-group if and only if every subgroup of Fit(G) is ??-semipermutable. A local version of this result is established in this paper. A subgroup H of G is said to be ??-seminormal provided that it is normalized by all Sylow subgroups Q such that (|H|, |Q|) =?1 and (|H|, |Q ^G |) ???1. It is shown that a finite solvable group is a PST-group if and only if every subgroup of Fit(G) is ??-seminormal in G.     

The least eigenvalue of graphs with given connectivity

Let G be a simple graph and A(G) be the adjacency matrix of G. The eigenvalues of G are those of A(G). In this paper, we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity.     

Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition (A,B,M), where A∼M and B⁄∼M, such that G[A]∈P, G[B]∈Q and G[M] is not (P,Q)-split. Then the graph G is reconstructible.     

Some properties of the spectrum of graphs

Let G be a graph and denote by Q(G)=D(G) A(G),L(G)=D(G)-A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. In this paper,some properties of the matrix Q(G)are studied. At the same time,anecessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given.     

QUANDLE VARIETIES,GENERALIZED SYMMETRIC SPACES,AND <Emphasis Type="Italic">φ</Emphasis>-SPACES

We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ?. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry.Assume that Q is normal as an algebraic variety and that the action of its inner automorphism group Inn(Q) has a dense orbit. Then we show that there is an algebraic group G acting on Q with the same orbits as Inn(Q) such that each G-orbit is isomorphic to the quandle (G/H, ?φ) associated to the group G, an automorphism φ of G and a subgroup H of Gφ.     

On Orbits in Double Flag Varieties for Symmetric Pairs

Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ? G or Q ? K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.     

Rational matrix groups of a special type

Let G be a finite group having a faithful irreducible character χ for which χ(1) is prime to ¦G¦/χ(1). Let n=[$\text{Q}$(χ):$\text{Q}$]χ(1), and assume that the factors are not both even. Then G can be embedded in GL_n($\text{Q}$) in such a way that its normalizer therein splits over its centralizer.