The maximal \alpha-index of trees with k pendent vertices and its computation

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. The $\alpha-$ index of $G$ is the spectral radius $\rho_{\alpha}\left( G\right)$ of the matrix $A_{\alpha}\left( G\right)=\alpha D\left( G\right) +(1-\alpha)A\left( G\right)$ where $\alpha \in [0,1]$. Let $T_{n,k}$ be the tree of order $n$ and $k$ pendent vertices obtained from a star $K_{1,k}$ and $k$ pendent paths of almost equal lengths attached to different pendent vertices of $K_{1,k}$. It is shown that if $\alpha\in\left[ 0,1\right) $ and $T$ is a tree of order $n$ with $k$ pendent vertices then% \[ \rho_{\alpha}(T)\leq\rho_{\alpha}(T_{n,k}), \] with equality holding if and only if $T=T_{n,k}$. This result generalizes a theorem of Wu, Xiao and Hong \cite{WXH05} in which the result is proved for the adjacency matrix ($\alpha=0$). Let $q=[\frac{n-1}{k}]$ and $n-1=kq+r$, $0 \leq r \leq k-1$. It is also obtained that the spectrum of $A_{\alpha}(T_{n,k})$ is the union of the spectra of two special symmetric tridiagonal matrices of order $q$ and $q+1$ when $r=0$ or the union of the spectra of three special symmetric tridiagonal matrices of order $q$, $q+1$ and $2q+2$ when $r \neq 0$. Thus the $\alpha-$ index of $T_{n,k}$ can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order $q+1$ if $r=0$ or order $2q+2$ if $r\neq 0$.

Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

Let $m$ be a positive integer whose smallest prime divisor is denoted by $p$, and let ${\Bbb Z}_m$ denote the cyclic group of residues modulo $m$. For a set $B=\{x_1,x_2,\ldots,x_m\}$ of $m$ integers satisfying $x_1 < x_2 < \cdots < x_m$, and an integer $j$ satisfying $2\leq j \leq m$, define $g_j(B)=x_j-x_1$. Furthermore, define $f_j(m,2)$ (define $f_j(m,{\Bbb Z}_m)$) to be the least integer $N$ such that for every coloring $\Delta : \{1,\ldots,N\}\rightarrow \{0,1\}$ (every coloring $\Delta : \{1,\ldots,N\}\rightarrow {\Bbb Z}_m$), there exist two $m$-sets $B_1,B_2\subset \{1,\ldots,N\}$ satisfying: (i) $\max(B_1) < \min(B_2)$, (ii) $g_j(B_1)\leq g_j(B_2)$, and (iii) $|\Delta (B_i)|=1$ for $i=1,2$ (and (iii) $\sum_{x\in B_i}\Delta (x)=0$ for $i=1,2$). We prove that $f_j(m,2)\leq 5m-3$ for all $j$, with equality holding for $j=m$, and that $f_j(m,{\Bbb Z}_m)\leq 8m+{m\over p}-6$. Moreover, we show that $f_j(m,2)\ge 4m-2+(j-1)k$, where $k=\left\lfloor\left(-1+\sqrt{{8m-9+j\over j-1}}\right){/2}\right\rfloor$, and, if $m$ is prime or $j\geq{m\over p}+p-1$, that $f_j(m,{\Bbb Z}_m)\leq 6m-4$. We conclude by showing $f_{m-1}(m,2)=f_{m-1}(m,{\Bbb Z}_m)$ for $m\geq 9$.

Optimal coercivity inequalities in W1,p(Ω)

This paper describes the characterization of optimal constants for some coercivity inequalities in W1,p(Ω), 1 < p < ∞. A general result involving inequalities of p-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on W1,p(Ω). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.

Optimal coercivity inequalities in W1,p(Ω)

This paper describes the characterization of optimal constants for some coercivity inequalities in W1,p(Ω), 1 < p < ∞. A general result involving inequalities of p-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on W1,p(Ω). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.

Note on analytic spread and asymptotic sequences

Since the foundational paper (10) by Northcott and Rees in 1954 there have been quite a few papers concerning reductions of ideals and the analytic spread of an ideal. One particular line of investigation concerning the analytic spread l(I) of an ideal I in a local ring (R, M) was begun in 1972 by Burch in (5), where it was shown that l(I) ≤ altitude R – min (grade R/In; n ≥ 1). This result was sharpened in 1980–81 by Brodmann in three papers, (2, 3, 4). Therein he showed that the sets {grade R/In; n ≥ 1} and {grade In−1/In; n ≥ 1} stabilize for all large n, and calling the stable values t and t*, respectively, it holds that t ≤ t* and l(I) ≤ altitude R – t* when I is not nilpotent. He then gave a case (involving R being quasi-unmixed) when equality holds. In 1981 in (20) Rees used two new approaches to Burch's inequality, and he proved two nice results which may both be stated as: l(I) ≤ altitude R – s(I) with equality holding when R is quasi-unmixed; here, s(I) = min {height P; P is a minimal prime divisor of (M, u) R[tI, u]}– 1 (in the first theorem), and s(I) is the length of a maximal asymptotic sequence over I (in the second theorem).

On Polynomials with Curved Majorants

A well-known result of Chebyshev is that if pn ∊ Pn, (Pn is the set of polynomials of degree at most n) and(1)then an(pn), the leading coefficient of pn, satisfies(2)with equality holding only for pn = ±Tn, where Tn is the Chebyshev polynomial of degree n. (See [6, p. 57].) This is an example of an extremal problem in which the norm of a given linear operator on Pn is sought. Another example is A. A. Markov's result that (1) implies that(3)There are also results for the linear functionals pn(k)(x0), x0 real, k = 1, … n – 1 ([8]).Suppose φ(x) ≧ 0 on [–1, 1] and (1) is generalized toas suggested by Rahman [4] (polynomials with curved majorants), what can then be said about the analogue of (3) or similar extremal problems?

The Isoperimetric Inequality for Curves with Self-Intersections

Banchoff and Pohl [3] have proved the following generalization of the isoperimetric inequality.Theorem. If γ is a closed, not necessarily simple, planar curve of length L, and w(p) is the winding number of a variable point p with respect to γ, then1with equality holding if and only if γ is a circle traversed a finite number of times in the same sense.

Families of finite sets satisfying an intersection condition

The following theorem is proved.Let X be a finite set of cardinality n ≥ 2, and let F be a family of subsets of X. Suppose that for F1, F2, F3 ∈ F we have |F1 ∩ F2 ∩ F3| ≥ 2. Then |F| ≤ 2n−2with equality holding if and only if for two different elements x, y of X, F = {F ⊆ X | x ∈ F, y ∈ F}.

Path decompositions of digraphs

A path decomposition of a digraph G (having no loops or multiple arcs) is a family of simple paths such that every arc of G lies on precisely one of the paths of the family. The path number, pn(G) is the minimal number of paths necessary to form a path decomposition of G.We show that pn(G) ≥ max{0, od(v)-id(v)} the sum taken over all vertices v of G, with equality holding if G is acyclic. If G is a subgraph of a tournament on n vertices we show that pn(G) ≤ with equality holding if G is transitive.We conjecture that pn(G) ≤ for any digraph G on n vertices if n is sufficiently large, perhaps for all n ≥ 4.

On Point-Symmetric Tournaments

A tournament is a directed graph in which there is exactly one arc between any two distinct vertices. Let denote the automorphism group of T. A tournament T is said to be point-symmetric if acts transitively on the vertices of T. Let g(n) be the maximum value of taken over all tournaments of order n. In [3] Goldberg and Moon conjectured that with equality holding if and only if n is a power of 3. The case of point-symmetric tournaments is what prevented them from proving their conjecture. In [2] the conjecture was proved through the use of a lengthy combinatorial argument involving the structure of point-symmetric tournaments. The results in this paper are an outgrowth of an attempt to characterize point-symmetric tournaments so as to simplify the proof employed in [2].