# Browse

We consider a graph whose vertices are legally colored using *k* colors and ask if the graph contains a *k*-clique. As it turns out this very special type of *k*-clique problem is in an intimate connection with constructing schedules. The practicality this clique search based construction of schedules is checked by carrying out numerical experiments.

Assume that *A _{j}
*,

*j*∈ {1, … ,

*m*} are positive definite matrices of order

*n*. In this paper we prove among others that, if 0 <

*l I*≤

_{n}*A*,

_{j}*j*∈ {1, … ,

*m*} in the operator order, for some positive constant

*l*, and

*I*is the unity matrix of order

_{n}*n*, then

where *Pk* ≥ 0 for *k* ϵ {1, …, *m*} and

The evolute of a conic in the pseudo-Euclidean plane is the locus of centers of all its osculating circles. It’s a curve of order six and class four in general case. In this paper we discuss and compute the order and class of evolutes of different types of conics. We will highlight those cases that have no analogy in the Euclidean plane.

Fast [12] is credited with pioneering the field of statistical convergence. This topic has been researched in many spaces such as topological spaces, cone metric spaces, and so on (see, for example [19, 21]). A cone metric space was proposed by Huang and Zhang [17]. The primary distinction between a cone metric and a metric is that a cone metric is valued in an ordered Banach space. Li et al. [21] investigated the definitions of statistical convergence and statistical boundedness of a sequence in a cone metric space. Recently, Sakaoğlu and Yurdakadim [29] have introduced the concepts of quasi-statistical convergence. The notion of quasi I-statistical convergence for triple and multiple index sequences in cone metric spaces on topological vector spaces is introduced in this study, and we also examine certain theorems connected to quasi I-statistically convergent multiple sequences. Finally, we will provide some findings based on these theorems.

For a graph *G*, we define the lower bipartite number LB(*G*) as the minimum order of a maximal induced bipartite subgraph of *G*. We study the parameter, and the related parameter bipartite domination, providing bounds both in general graphs and in some graph families. For example, we show that there are arbitrarily large 4-connected planar graphs *G* with LB(*G*) = 4 but a 5-connected planar graph has linear LB(*G*). We also show that if *G* is a maximal outerplanar graph of order *n*, then LB(*G*) lies between (*n* + 2)/3 and 2 *n*/3, and these bounds are sharp.

We study the path behavior of the symmetric walk on some special comb-type subsets of ℤ^{2} which are obtained from ℤ^{2} by generalizing the comb having finitely many horizontal lines instead of one.

In a typical maximum clique search algorithm when optimality testing is inconclusive a forking takes place. The instance is divided into smaller ones. This is the branching step of the procedure. In order to ensure a balanced work load for the processors for parallel algorithms it is essential that the resulting smaller problems are do not overly vary in difficulty. The so-called splitting partitions of the nodes of the given graph were introduced earlier to meliorate this problem. The paper proposes a splitting partition of the edges for the same purpose. In the lack of available theoretical tools we assess the practical feasibility of constructing suboptimal splitting edge partitions by carrying out numerical experiments. While working with splitting partitions we have realized that they can be utilized as preconditioning tools preliminary to a large scale clique search. The paper will discuss this new found role of the splitting edge partitions as well.

In this paper, we introduce the notion of a Gel’fand Γ-semiring and discuss the various characterization of simple, *k*-ideal, strong ideal, *t*-small elements and additively cancellative elements of a Gel’fand Γ-semiring *R*, and prove that the set of additively cancellative elements, set of all *t*-small elements of *R* and set of all maximal ideal of *R* are strong ideals. Further, let *R* be a simple Gel’fand Γ-semiring and 1 ≠ *t* ∈ *R*. Let *M* be the set of all maximal left (right) ideals of *R*. Then an element *x* of *R* is *t*-small if and only if it belongs to every maximal one sided left (right)ideal of *R* containing *t*.

For a continuous and positive function *w*(λ), *λ >
* 0 and

*μ*a positive measure on (0, ∞) we consider the following

*integral transform*

where the integral is assumed to exist for *t* > 0.

We show among others that *D*(*w, μ*) is operator convex on (0, ∞). From this we derive that, if *f* : [0, ∞) → **R** is an operator monotone function on [0, ∞), then the function [*f*(0) -*f*(*t*)] *t*
^{-1} is operator convex on (0, ∞). Also, if *f* : [0, ∞) → **R** is an operator convex function on [0, ∞), then the function

under some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.

Problem 2 of Welsh’s 1976 text *Matroid Theory*, asking for criteria telling when two families of sets have a common transversal, is solved.

Another unsolved problem in the text *Matroid Theory*, on whether the “join” of two non-decreasing submodular functions is submodular, is answered in the negative. This resolves an issue first raised by Pym and Perfect in 1970.