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In 2009, Kyaw proved that every n-vertex connected K 1,4 -free graph G with σ 4 (G) ≥ n − 1 contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw's result for connected K 1,5 -free graphs. We show that every n-vertex connected K 1,5 -free graph G with σ 5 (G) ≥ n − 1 contains a spanning tree with at most 4 leaves. Moreover, the degree sum condition "σ 5 (G) ≥ n − 1" is best possible.

Let T be a tree, a vertex of 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(T ) and the set of branch vertices of T is denoted by B(T ). 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(TThe resulting graph is a subtree of T and is denoted by R Stem(T ). It is called the reducible stem of T . A leaf of R Stem(T ) is called a peripheral branch vertex of T . In this paper, we give some sharp sufficient conditions on the independence number and the degree sum for a graph G to have a spanning tree with few peripheral branch vertices.

A braided Ann-category A is an Ann-category A together with the braiding c such that (A, ⊗, a, c, (1, l, r)) is a braided tensor category, and c is compatible with the distributivity constraints. The paper shows the dependence of the left (or right) distributivity constraint on other axioms. Hence, the paper shows the relation to the concepts of distributivity category due to M. L. Laplaza and ring-like category due to A. Fröhlich and C.T.C Wall.The center construction of an almost strict Ann-category is an example of a nonsymmetric braided Ann-category.

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