The rank of Pseudo walk matrices: controllable and recalcitrant pairs
ODAM-Vol. 3 (2020), Issue 3, pp. 41 – 52 Open Access Full-Text PDF
Alexander Farrugia
Abstract: A pseudo walk matrix \(\bf{W}_\bf{v}\) of a graph \(G\) having adjacency matrix \(\bf{A}\) is an \(n\times n\) matrix with columns \(\bf{v},\bf{A}\bf{v},\bf{A}^2\bf{v},\ldots,\bf{A}^{n-1}\bf{v}\) whose Gram matrix has constant skew diagonals, each containing walk enumerations in \(G\). We consider the factorization over \(\mathbb{Q}\) of the minimal polynomial \(m(G,x)\) of \(\bf{A}\). We prove that the rank of \(\bf{W}_\bf{v}\), for any walk vector \(\bf{v}\), is equal to the sum of the degrees of some, or all, of the polynomial factors of \(m(G,x)\). For some adjacency matrix \(\bf{A}\) and a walk vector \(\bf{v}\), the pair \((\bf{A},\bf{v})\) is controllable if \(\bf{W}_\bf{v}\) has full rank. We show that for graphs having an irreducible characteristic polynomial over \(\mathbb{Q}\), the pair \((\bf{A},\bf{v})\) is controllable for any walk vector \(\bf{v}\). We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors \(\bf{v}\), the degree of the minimal polynomial of the largest eigenvalue of \(\bf{A}\) is a lower bound for the rank of \(\bf{W}_\bf{v}\). If the rank of \(\bf{W}_\bf{v}\) attains this lower bound, then \((\bf{A},\bf{v})\) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that \((\bf{A},\bf{v})\) is neither controllable nor recalcitrant for any walk vector \(\bf{v}\).
Uniformity of dynamic inequalities constituted on time Scales
EASL-Vol. 3 (2020), Issue 4, pp. 19 – 27 Open Access Full-Text PDF
Muhammad Jibril Shahab Sahir
Abstract: In this article, we present extensions of some well-known inequalities such as Young’s inequality and Qi’s inequality on fractional calculus of time scales. To find generalizations of such types of dynamic inequalities, we apply the time scale Riemann-Liouville type fractional integrals. We investigate dynamic inequalities on delta calculus and their symmetric nabla results. The theory of time scales is utilized to combine versions in one comprehensive form. The calculus of time scales unifies and extends some continuous forms and their discrete and quantum inequalities. By applying the calculus of time scales, results can be generated in more general form. This hybrid theory is also extensively practiced on dynamic inequalities.
On divisible and pure multigroups and their properties
OMS-Vol. 4 (2020), Issue 1, pp. 377 – 385 Open Access Full-Text PDF
P. A. Ejegwa, M. A. Ibrahim
Abstract: The theory of multigroups is a generalized group’s theoretic notions in multiset framework. Although myriad of researches have been done in multigroup theory, but some group’s analogue concepts have not been investigated in multigroup setting. In this paper we propose the notions of divisible and pure multigroups and characterize some of their properties. It is established that the image and preimage of homomorphism of divisible and pure multigroups are divisible and pure multigroups. The nexus between divisible and pure multigroups and that of divisible and pure groups are instituted using the concept of cuts of multigroups.
Coupled coincidence and coupled common fixed point theorem in dislocated quasi metric space
EASL-Vol. 3 (2020), Issue 4, pp. 11 – 18 Open Access Full-Text PDF
Mitiku Damene, Kidane Koyas, Solomon Gebregiorgis
Abstract: The objective of this paper is to establish a theorem involving a pair of weakly compatible mappings fulfilling a contractive condition of rational type in the context of dislocated quasi metric space. Besides we proved the existence and uniqueness of coupled coincidence and coupled common fixed point for such mappings. This work offers extension as well as considerable improvement of some results in the existing literature. Lastly, an illustrative example is given to validate our newly proved results.
On hyper-singular integrals
OMA-Vol. 4 (2020), Issue 2, pp. 100 – 103 Open Access Full-Text PDF
Alexander G. Ramm
Abstract: The integrals \(\int_{-\infty}^\infty t_+^{\lambda-1} \phi(t)dt\) and \(\int_0^t(t-s)^{\lambda -1}b(s)ds\) are considered, \(\lambda\neq 0,-1,-2…\), where \(\phi\in C^\infty_0(\mathbb{R})\) and \(0\le b(s)\in L^2_{loc}(\mathbb{R})\). These integrals are defined in this paper for \(\lambda\le 0\), \(\lambda\neq 0,-1,-2,…\), although they diverge classically for \(\lambda\le 0\). Integral equations and inequalities are considered with the kernel \((t-s)^{\lambda -1}_+\).
On odd prime labeling of graphs
ODAM-Vol. 3 (2020), Issue 3, pp. 33 – 40 Open Access Full-Text PDF
Maged Zakaria Youssef, Zainab Saad Almoreed
Abstract: In this paper we give a new variation of the prime labeling. We call a graph \(G\) with vertex set \(V(G)\) has an odd prime labeling if its vertices can be labeled distinctly from the set \(\big\{1, 3, 5, …,2\big|V(G)\big| -1\big\}\) such that for every edge \(xy\) of \(E(G)\) the labels assigned to the vertices of \(x\) and \(y\) are relatively prime. A graph that admits an odd prime labeling is called an odd prime graph. We give some families of odd prime graphs and give some necessary conditions for a graph to be odd prime. Finally, we conjecture that every prime graph is odd prime graph.
