We use graphs to define sets of Salem and Pisot numbers, and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers n the smallest known element of the n-th derived set of the set of Pisot numbers comes from a graph. We define the Mahler measur...

A delivery person must leave the central location of the business, deliver packages at a number of addresses, and then return. Naturally, he/she wishes to reduce costs by finding the most efficient route. This motivates the following: Given a set of k distinct vertices S = {x1, x2, . . . , xk} in a simple graph G, the closed k-stop-distance of set S is defined to be dk(S) = min θ∈P(S) (d(θ(x1),...

In this notes unbounded regular operators on Hilbert C∗-modules over arbitrary C∗-algebras are discussed. A densely defined closed operator t possesses a densely defined adjoint operator if the graph of t is an orthogonal summand. Moreover, for a densely defined closed operator t the graph of t is orthogonally complemented if and only if t is regular. For a given C*-algebra A any densely define...

Let G = (V,E) be a graph of order n. A closed distance magic labeling of G is a bijection l : V (G) → {1, . . . , n} for which there exists a positive integer k such that ∑ x∈N [v] l(x) = k for all v ∈ V , where N [v] is the closed neighborhood of v. We consider the closed distance magic graphs in the algebraic context. In particular we analyze the relations between the closed distance magic la...

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed g and k, we show that every n-vertex graph that can be embedded on a surface of genus g with at most k cro...

In recent years, graph mining has attracted much attention in the data mining community. Several efficient frequent subgraph mining algorithms have been recently proposed. However, the number of frequent graph patterns generated by these graph mining algorithms may be too large to be effectively explored by users, especially when the support threshold is low. In this paper, we propose to summar...