Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower-${\ell }_q$-tree estimate and let T be a bounded linear operator from X which satisfies an upper-${\ell }_p$-tree estimate. Then T factors through a subspace of ${\left(\sum Fₙ\right)}_{{\ell }_r}$, where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $({\ell }_p,{\ell }_q)$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic...

For a connected graph $G=(V,E)$ and a set $S\subseteq V\left(G\right)$ with at least two vertices, an $S$-Steiner tree is a subgraph $T=(V^{\text{'}},E^{\text{'}})$ of $G$ that is a tree with $S\subseteq V^{\text{'}}$. If the degree of each vertex of $S$ in $T$ is equal to 1, then $T$ is called a pendant $S$-Steiner tree. Two $S$-Steiner trees are if they share no vertices other than $S$ and have no edges in common. For $S\subseteq V\left(G\right)$ and $\left|S\right|\ge 2$, the pendant tree-connectivity ${\tau }_G\left(S\right)$ is the maximum number of internally disjoint pendant $S$-Steiner trees in $G$, and for $k\ge 2$, the $k$-pendant tree-connectivity ${\tau }_k\left(G\right)$ is the...

Let $L(n,d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d.$ Lesniak (1975) gave the lower bound $B(n,d)=\lceil 2(n-1)/d\rceil$ for $L(n,d).$ When $d$ is even, $B(n,d)=L(n,d).$ But when $d$ is odd, $B(n,d)$ is smaller than $L(n,d)$ in general. For example, $B(21,3)=14$ while $L(21,3)=19.$ In this note, we determine $L(n,d)$ using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.

A graph $G$ is a $k$-tree if either $G$ is the complete graph on $k+1$ vertices, or $G$ has a vertex $v$ whose neighborhood is a clique of order $k$ and the graph obtained by removing $v$ from $G$ is also a $k$-tree. Clearly, a $k$-tree has at least $k+1$ vertices, and $G$ is a 1-tree (usual tree) if and only if it is a $1$-connected graph and has no $K_3$-minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of $k$-trees as follows: if $G$ is a graph with at least $k+1$ vertices, then $G$ is...

One way to generalize complete Erdős space ${}_c$ is to consider uncountable products of zero-dimensional $G_{\delta }$-subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise...

The Sombor index $SO\left(G\right)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_G^2\left(u\right)+d_G^2\left(v\right)}$ of all edges $uv$ of $G$, where $d_G\left(u\right)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G=(V,E)$ is called a quasi-tree if there exists $u\in V\left(G\right)$ such that $G-u$ is a tree. Denote $𝒬(n,k)=\{G:G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G\left(u\right)=k\}$. We determined the minimum and the second minimum Sombor indices of all quasi-trees in $𝒬(n,k)$. Furthermore, we characterized the corresponding extremal graphs, respectively.

Let $T$ be a tree. Then a vertex of $T$ with degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. The set of leaves of $T$ is denoted by $L\left(T\right)$ and the set of branch vertices of $T$ is denoted by $B\left(T\right)$. For two distinct vertices $u$, $v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v.$ Let $T$ be a tree with $B\left(T\right)\ne \varnothing$. For each leaf $x$ of $T$, let $y_x$ denote the nearest branch vertex to $x$. We delete $V\left(P_T[x,y_x]\right)\setminus \left\{y_x\right\}$ from $T$ for all $x\in L\left(T\right)$. The resulting subtree of $T$ is called the reducible stem...

A proper coloring $c:V\left(G\right)\to \{1,2,...,k\}$, $k\ge 2$ of a graph $G$ is called a graceful $k$-coloring if the induced edge coloring $c^{\text{'}}:E\left(G\right)\to \{1,2,...,k-1\}$ defined by $c^{\text{'}}\left(uv\right)=|c\left(u\right)-c\left(v\right)|$ for each edge $uv$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$-coloring is the graceful chromatic number ${\chi }_g\left(G\right)$. It is known that if $T$ is a tree with maximum degree $\Delta$, then ${\chi }_g\left(T\right)\le \lceil \frac{5}{3}\Delta \rceil$ and this bound is best possible. It is shown for each integer $\Delta \ge 2$ that there is an infinite class of trees $T$ with maximum degree $\Delta$ such that ${\chi }_g\left(T\right)=\lceil \frac{5}{3}\Delta \rceil$. In particular, we investigate for each...

The topology and combinatorial structure of the Mandelbrot set ${ℳ}^d$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in ${ℳ}^d$. Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, ${\Lambda }^d$. In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence...

By using the concepts of limited $p$-converging operators between two Banach spaces $X$ and $Y$, $L_p$-sets and $L_p$-limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$-Dunford–Pettis property of order $p$ and Pelczyński’s property of order $p$, $1\le p<\infty$.

We study weakly precompact sets and operators. We show that an operator is weakly precompact if and only if its adjoint is pseudo weakly compact. We study Banach spaces with the $p$-$L$-limited${}^{*}$ and the $p$-(SR${}^{*}$) properties and characterize these classes of Banach spaces in terms of $p$-$L$-limited${}^{*}$ and $p$-Right${}^{*}$ subsets. The $p$-$L$-limited${}^{*}$ property is studied in some spaces of operators.

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

We show that for a linear space of operators $ℳ\subseteq ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ the following assertions are equivalent. (i) $ℳ$ is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map $\Psi =({\psi }_1,{\psi }_2)$ on a bilattice $\mathrm{Bil}\left(ℳ\right)$ of subspaces determined by $ℳ$ with $P\le {\psi }_1(P,Q)$ and $Q\le {\psi }_2(P,Q)$ for any pair $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$, and such that an operator $T\in ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ lies in $ℳ$ if and only if ${\psi }_2(P,Q)T{\psi }_1(P,Q)=0$ for all $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$. This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

Equivalent formulations of the Dunford-Pettis property of order $p$ (${DPP}_p$), $1<p<\infty$, are studied. Let $L(X,Y)$, $W(X,Y)$, $K(X,Y)$, $U(X,Y)$, and $C_p(X,Y)$ denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and $p$-convergent operators from $X$ to $Y$. Classical results of Kalton are used to study the complementability of the spaces $W(X,Y)$ and $K(X,Y)$ in the space $C_p(X,Y)$, and of $C_p(X,Y)$ in $U(X,Y)$ and $L(X,Y)$.

Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is $val\left(f\right)={\Sigma }_{e\in E\left(G\right)}{f}^{\text{'}}K\left(e\right)$. The maximum value of a γ-labeling of G is defined as $va{l}_{max}\left(G\right)=maxval\left(f\right):fisa\gamma -labelingofG$; while the minimum value of a γ-labeling of G is $va{l}_{min}\left(G\right)=minval\left(f\right):fisa\gamma -labelingofG$; The values $va{l}_{max}\left(S_{p,q}\right)$ and $va{l}_{min}\left(S_{p,q}\right)$ are determined for double stars $S_{p,q}$. We present characterizations of connected graphs G of order n for which...