Releasing the exact degree sequence of a graph for analysis may violate privacy. However, the degree sequence of a graph is an important summary statistic that is used in many statistical models. Hence a natural starting point is to release a private version of the degree sequence. A graphical degree partition is a monotonic degree sequence such that there exists a simple graph realizing the se...

In this paper existence and uniqueness of twelfth degree spline is proved with application to quadrature. This formula is in the class of splines of degree 12 and continuity order [Formula: see text] that matches the derivatives up to order 6 at the knots of a uniform partition. Some mistakes in the literature are pointed out and corrected. Numerical examples are given to illustrate the applica...

Let G = (V, E) be a graph. A subset D of V is called common neighbourhood dominating set (CN-dominating set) if for every v ∈ V −D there exists a vertex u ∈ D such that uv ∈ E(G) and |Γ(u, v)| > 1, where |Γ(u, v)| is the number of common neighbourhood between the vertices u and v. The minimum cardinality of such CN-dominating set denoted by γcn(G) and is called common neighbourhood domination n...

Equitable gender representation is an important aspect of scientific workforce development to secure a sufficient number of individuals and a diversity of perspectives. Biology is the most gender equitable of all scientific fields by the marker of degree attainment, with 52.5% of PhDs awarded to women. However, equitable rates of degree completion do not translate into equitable attainment of f...

An equitable k-colouring of a graph G is a proper k-colouring, for which any two colour classes differ in size by at most one. Equitable colourings naturally arise in some scheduling, partitioning, and load-balancing problems [1, 15, 16]. Pemmaraju [13] and Janson and Ruciński [6] used equitable colourings to derive deviation bounds for sums of random variables that exhibit limited dependence. ...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {±1,±2, . . . ,±k} is said to be a signed star {k}-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. The signed star {k}-domination number of a graph G is γ{k}SS(G) = min{ ∑ e∈E f(e) | f is a S...

We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular code (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the percentage of the colors over the code. For some partial cases of completely regular codes we derive explicit formulas of weig...

We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular code (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the percentage of the colors over the code. For some partial cases of completely regular codes we derive explicit formulas of weig...