Characterizing Convergence Conditions for the Mα-Integral

Park, Ryu, and Lee recently defined a Henstock-type integral, which lies entirely between the McShane and the Henstock integrals. This paper presents two characterizing convergence conditions for this integral, and derives other known convergence theorems as corollaries

On Completely k-Magic Regular Graphs

Let k be a positive integer. A graph G = (V (G), E(G)) is said to be k-magic if there is a function (or edge labeling) ` : E(G) → Zk \ {0}, where Z1 = Z, such that the induced function (or vertex labeling) ` + : V (G) → Zk, defined by ` +(v) = P uv∈E(G) `(uv), is a constant map, where the sum is taken in Zk. We say that G is c-sum k-magic if ` +(v) = c for all v ∈ V (G). The set of all c ∈ Zk such that G is c-sum k-magic is called the sum spectrum of G with respect to k. In the case when the sum spectrum of G is Zk, we say that G is completely k-magic. In this paper, we determine all completely 1-magic regular graphs. After observing that any 2-magic graph is not completely 2-magic, we show that some regular graphs are completely k-magic for k ≥ 3, and determine the sum spectra of some regular graphs that are not completely k-magic

Cauchy Extension of Mα-Integral and Absolute Mα-Integrability

Park, Ryu, and Lee recently introduced a Henstock-type integral, which lies between the Mcshane and the Henstock integrals. This paper proves the closure property of this new integral under Cauchy extension, and presents a characterization on absolute Mα-integrabilit

Characterization of completely k-magic regular graphs

Let k ∈ N and c ∈ Zk. A graph G is said to be c-sum k-magic if there is a labeling ` : E(G) → Zk \ {0} such that P u∈N(v) `(uv) ≡ c (mod k) for every vertex v of G, where N(v) is the neighborhood of v in G. We say that G is completely k-magic whenever it is c-sum k-magic for every c ∈ Zk. In this paper, we characterize all completely k-magic regular graphs

Computing the metric dimension of truncated wheels

For an ordered subset W = {w1, w2, w3, . . . , wk} of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v|W) = (d(v, w1), d(v, w2), d(v, w3), . . . , d(v, wk)). The set W is called a resolving set of G if r(u|W) = r(v|W) implies u = v. The metric dimension of G, denoted by β(G), is the minimum cardinality of a resolving set of G. Let n ≥ 3 be an integer. A truncated wheel, denoted by TWn, is the graph with vertex set V (TWn) = {a} ∪ B ∪ C, where B = {bi : 1 ≤ i ≤ n} and C = {cj,k : 1 ≤ j ≤ n, 1 ≤ k ≤ 2}, and edge set E(TWn) = {abi : 1 ≤ i ≤ n} ∪ {bici,k : 1 ≤ i ≤ n, 1 ≤ k ≤ 2} ∪ {cj,1cj,2 : 1 ≤ j ≤ n} ∪ {cj,2cj+1,1 : 1 ≤ j ≤ n}, where cn+1,1 = c1,1. In this paper, we compute the metric dimension of truncated wheels

Revisiting a Number-Theoretic Puzzle: The Census-Taker Problem

The current work revisits the results of L.F. Meyers and R. See in [3], and presents the census-taker problem as a motivation to introduce the beautiful theory of numbers

Strong Derivative and the Essentially Riemann Integral

We characterize the primitives of Riemann integrable functions using the concept of strong differentiability. We also introduce the notion of essentially Riemann integrable function and characterize it descriptively using its primitive. We also give a characterization of this function that parallels Lebesgue\u27s criterion for Riemann integrability

Characterizing 2-distance graphs

Let X be a finite simple graph. The 2-distance graph D2(X) of X is the graph with the same vertex set as X and two vertices are adjacent if and only if their distance in X is exactly 2. A graph G is a 2-distance graph if there exists a graph X such that D2(X)≅G. In this paper, we give three characterizations of 2-distance graphs, and find all graphs X such that D2(X)≅kP2 or Km∪Kn, where k≥2 is an integer, P2 is the path of order 2, and Km is the complete graph of order m≥1