the order graph of a group $g$, denoted by $gamma^*(g)$, is a graph whose vertices are subgroups of $g$ and two distinct vertices $h$ and $k$ are adjacent if and only if $|h|big{|}|k|$ or $|k|big{|}|h|$. in this paper, we study the connectivity and diameter of this  graph. also we give a relation between the order graph and prime  graph of a group.

let f, g and h be non-empty graphs. the notation f → (g,h) means that if any edge of f is colored by red or blue, then either the red subgraph of f con- tains a graph g or the blue subgraph of f contains a graph h. a graph f (without isolated vertices) is called a ramsey (g,h)−minimal if f → (g,h) and for every e ∈ e(f), (f − e) 9 (g,h). the set of all ramsey (g,h)−minimal graphs is denoted by ...

in this paper, we first collect the earlier results about some graph operations and then wepresent applications of these results in working with chemical graphs.

The spectral theorem for commuting self-adjoint operators along with the associated functional (or operational) calculus is among the most useful and beautiful results of analysis. It is well known that forming a functional calculus for noncommuting self-adjoint operators is far more problematic. The central result of this paper establishes a rich functional calculus for any finite number of no...

Polynomials, rational functions, and formal power series in (free) noncommuting variables were considered in a variety of settings. While usually viewed as formal algebraic objects, they also appeared often as functions by substituting tuples of matrices or operators for the variables. Our point of view is that a function of noncommuting variables is a function defined on tuples of matrices of ...

Refer ences Al-Thagafi, M. A. , Common fixed points and best approximation, J. Approx Theory, 85, (1996), 318–323. Al-Thagafi, M. A. and Shahzad, N. , Noncommuting selfmaps and invariant approximations, Nonlinear Anal. , 64, (2006), 2778–2786. Al-Thagafi, M. A. and Shahzad, N. , Generalized I-nonexpansive selfmaps and invariant approximations, Acta Math Sinica, 24, (2008), 867–876. Beg, I. , Sh...

let g be a group. a subset x of g is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in x. if |x| ≥ |y | for any other set of pairwise non-commuting elements y in g, then x is said to be a maximal subset of pairwise non-commuting elements. in this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

The COVID-19 pandemic and resulting government-enforced lockdown affected the travel behavior lives of people worldwide. In this research, hierarchical cluster analysis (HCA) is used to quantify impact on daily flow profiles cyclists due public’s response different levels restrictions during a 6-month period in 2020. An inductive loop network Tyne Wear, UK provided cycle data from 25 sites. A p...

We introduce a remarkable new family of norms on the space $n \times n$ complex matrices. These arise from combinatorial properties symmetric functions, and their construction validation involve probability theory, partition combinatorics, trace polynomials in noncommuting variables. Our enjoy many desirable analytic algebraic properties, such as an elegant determinantal interpretation ability ...