Let T be a tree. A vertex of degree one is called a leaf. The set of leaves of T is denoted by L(T). The subtree T-L(T) of T is called the stem of T and denoted by stem(T) . In this paper, we give a sufficient condition of spectral radius of a graph to show that it has a spanning tree whose stem has few leaves.
In this article we extended the contraction principle to the complete ℂ^* -algebra-valued semimetric spaces that fulfill an extra regularity property. Next, we study some properties of quasisymmetric maps in ℂ^* -algebra-valued regular semimetric spaces. Using very specific hypotheses we prove that every quasisymmetric map of a ℂ^* -algebra-valued regular semimetric space admits at least one fixed point.
For a non-square positive integer d, it is well known that the continued fraction expansion of √(d) (respectively (1+√(d))/2 ) has the form [a_0; a_1,…, a_l_d-1, 2a_0] ( respectively [a'_0; a'_1,…, a'_l'_d-1, 2a'_0-1]), where a_1,…, a_l_d-1 (respectively a'_1,…, a'_l'_d-1 ) is a palindromic sequence of positive integers. For a given positive integer l (≥ 2) (respectively l' (≥ 2) ) and a palindromic sequence of positive integers a_1,…, a_l-1 (respectively a'_1,…, a'_l'-1 ), we define the set S(l;a_1,…, a_l-1):={d∈ℤ^+ | d is squarefree , √(d)=[a_0; a_1,…, a_l-1, 2a_0], where a'_0=⌊√(d)⌋} (respectively S'(l';a'_1, … , a'_l'-1) :={d∈ℤ^+ | d is squarefree , 1+√(d)/2=[a'_0; a'_1,…, a'_l'-1, 2a'_0-1], where a'_0=⌊1+√(d)/2⌋} . In this paper, we show that almost all square-free integers are the least elements of S(l;a_1,…, a_l-1) or S'(l';a'_1,…, a'_l'-1) .
A Steiner coset partition of a group G with respect to pairwise distinct subgroups H_1 , ..., H_r is a collection of pairwise disjoint cosets g_1H_1 , ..., g_rH_r whose union is G. Motivated by the Herzog–Schönheim Conjecture, we have recently started exploring the groups that admit Steiner coset partitions. In previous work, we completely classified such groups for up to r=4 mutually commuting subgroups. In this paper, we extend the classification to r=5 mutually commuting subgroups.
Using Fourier-analytic methods, we derive a Hardy-type oscillatory identity for the family of planar domains {x∈ℝ^2| |x_1|^p + |x_2|^p ≤ r^p} with positive p satisfying (2/p) ∈ℕ . This identity generalizes the classical Hardy’s identity for the Euclidean disk, arising from the Poisson summation formula. For these domains of non-smooth curves (including the astroid), the rotational symmetry of the Fourier transform is lost, so it cannot be expressed using classical Bessel functions. To address this, we introduce a family of generalized Bessel functions naturally arising from the Fourier transform of p-radial functions. These functions capture the oscillatory structure induced by the loss of rotational symmetry and provide the analytic framework for Hardy-type identities between these domains. While classical motivations relate to estimating lattice point errors in number theory, we focus here on deriving the identities themselves and understanding their structure.
