Asian-European Journal of Mathematics

The energy of a square matrix is the sum of the absolute values of deviations of its eigenvalues from their mean. In this paper, we extend the notion of the Laplacian and the signless Laplacian energy of a graph to non-negative real symmetric matrices with zero diagonal. Such a matrix $\widehat{A}$ can be considered as the adjacency matrix of some weighted graph. Considering the sum of the weights of edges that are incident to a vertex as the weight of the corresponding vertex, we construct the diagonal matrix $\widehat{D}$ of which the diagonal entries are the weights of the corresponding vertices. We define a weighted version of the Laplacian matrix as $\widehat{L}=\widehat{D}-\widehat{A}$ and that of the signless Laplacian matrix as $\widehat{Q}=\widehat{D}+\widehat{A}$. In this work, we obtain some bounds of the energies of the weighted adjacency matrix, the weighted Laplacian matrix, and the weighted signless Laplacian matrix by studying the corresponding weighted graph. We also obtain some relations interconnecting those energies.

Assume that $R$ is a commutative ring with nonzero identity. The comaximal graph of $R$, denoted by $\mathrm{\Gamma }(R)$, is a simple graph whose vertex set consists of all elements of $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. In this paper, we investigate the signless Laplacian spectrum of the comaximal graph of the ring ${\mathbb{Z}}_n$.

For a connected simple graph $G$ and a positive integer $k\le \mathrm{\text{diam}}(G)$, a radio $k$-coloring is an assignment $f$ of positive integers (colors) to the vertices of $G$ such that for every pair of distinct vertices $u$ and $v$ in $G$, $|f(u)-f(v)|\ge 1+k-d(u,v)$. The span $r{c}_k(f)$ of $f$ is ${max}_{v\in V(G)}f(v)$. The minimum of $\{r{c}_k(f):f$ is a radio $k$-coloring of $G\}$ is called the radio $k$-chromatic number of $G$ and is denoted by $r{c}_k(G)$. If $d$ is the diameter of $G$, then a radio $d$-coloring is referred as a radio coloring and the radio $d$-chromatic number as the radio number, $rn(G)$, of $G$. In this paper, we obtain an upper bound for the radio k-chromatic number of the corona of two graphs $G$ and $H$ for $k\ge 2$, and we obtain a necessary condition for this upper bound to be exact. Also, we see that for path $P_m$, $m$ even and complete graph $K_n$ the upper bound is sharp for $r{c}_k(K_n\odot H)$ and $rn(P_m\odot H)$, where $H$ is an arbitrary graph. Further, we give a lower bound and an improved upper bound for $rn(P_m\odot H)$, $m$ odd and $rn(Q_n\odot H)$, $Q_n$ is hypercube.

For $a,b\in ℝ$, the general eccentric distance sum of a connected graph $G$ is defined as ${\mathrm{\text{EDS}}}_{a,b}(G)={\sum }_{u\in V(G)}{[{\mathrm{\text{ecc}}}_G(u)]}^a{[D_G(u)]}^b$, where $V(G)$ is the vertex set of $G$, ${\mathrm{\text{ecc}}}_G(u)$ is the eccentricity of $u\in V(G)$, $D_G(u)={\sum }_{v\in V(G)}{d}_G(u,v)$ and $d_G(u,v)$ is the distance between vertices $u$ and $v$ in $G$. For $a\ge 0$ and $b\ge 1$, we present the graphs having the smallest general eccentric distance sum among graphs with given order and diameter, and among bipartite graphs with given order and odd diameter. The extremal graphs for the classical eccentric distance sum are corollaries of our results on the general eccentric distance sum.

In this paper, first, we give the general formulas according to first fundamental form of a surface for different types of loxodromes, meridians and surfaces in $E_1^3$. After that, we obtain the differential equations of loxodromes on Type-I, Type-II and Type-III twisted surfaces in $E_1^3$ and also, we state a theorem which generalizes the differential equations of different types of loxodromes on the twisted surfaces for a special case. Finally, we provide several examples for visualizing our obtained results and draw our loxodromes and meridians on twisted surfaces with the aid of Mathematica.

One decade ago, Beeler and Hoilman introduced the game of peg solitaire to arbitrary graphs. Since then, some progress has been made characterizing solvable graphs. Furthermore, several variants of the original game have been considered. The main goal of this paper is to give a proper overview of the results that have been obtained so far (until December 2021) as well as to show which methods are used to prove these. We also present important open questions and research directions, including some new results. The content should be helpful to mathematicians already interested in this topic, but, on the other hand, will provide a perfect starting point for (also early career) researchers looking for a fruitful topic in combinatorial game theory.

In this paper, we investigate Poison kernel associated with Kontorovich–Lebedev transform. We significantly simplify the integral yielding a tractable form which is more useful for explicit calculation. We establish also that Kontorovich–Lebedev multipliers of Laplace transform type are bounded from $L^p(w)$ into it self when $1<p<\infty ,$ and from $L^1(w)$ into $L^{1,\infty }(w),$ provided that $w$ is in the Muckenhoupt class $A^p$ on $((0,\infty ),dx)$. At the end, an integral representation of the operator ${\mathrm{\Delta }}^{-\beta /2}f$ is obtained and its boundedness has been discussed in Lebesgue space.

Let $p_1,p_2,\dots ,p_r,q$ be distinct primes, $q$ odd. Let $m={\prod }_{i=1}^r{p}_i$, where $r\ge 2$ be an integer. In this paper, it is observed that the explicit expressions of primitive idempotents from $R_{p_i}$ are sufficient to compute the explicit expressions of primitive idempotent in semisimple ring $R_m=F_q[x]/(x^m-1)$. It is also shown that the results obtained in [A. Sahni and P. T. Sehgal, Minimal cyclic codes of length $p^{n}q$, Finite Fields Appl. 18(5) (2012) 1017–1036; P. Kumar and S. K. Arora, $\lambda$-mapping and primitive idempotents in semisimple ring ${\Re }_m$, Comm. Algebra 41(10) (2013) 3679–3694] are simple corollaries to the results obtained in the paper.

In this paper, we study the concept of weakly almost periodic functions on Fréchet algebras. For a Fréchet algebra ${𝒜}$, we show that $\mathrm{\text{WAP}}({𝒜})=wap({𝒜})$. We also show that ${{𝒜}}^{**}$ is Arens regular if and only if both ${𝒜}$ and $\mathrm{\text{WAP}}{({𝒜})}^{*}$ are Arens regular. Finally, for a sequence of Fréchet algebras $({{𝒜}}_n)$, we prove that the Fréchet algebra ${\ell }^1\text{-}{\prod }_{n\in \mathbb{N}}{{𝒜}}_n$ is Arens regualr if and only if each ${{𝒜}}_n$ is Arens regular.

Jones monoids ${{𝒥}}_n$, for $1<n$, is a family of monoids relevant in knot theory. The purpose of this paper is to characterize the identities satisfied by the Jones monoid ${{𝒥}}_5$.

The harmonic series diverges. But, Kempner [A curious convergent series, Amer. Math Monthly 21 (1914) 48–50] proved that by removing those terms from the harmonic series whose denominators contain a digit $9$ anywhere, the resulting sub-series converges. Thus, Kempner allowed no 9’s at all, but in this short note, we allow 9’s with some restrictions and without tampering the convergence of the resultant series. Some results of our note generalize a recent note of Mukherjee and Sarkar [A short note on a curious convergent series, Asian-Eur. J. Math. 14(09) (2021) 2150158, doi:10.1142/S1793557121501588].

In this paper, we initiate the concept of matrix transformation through uncertain measure of a complex uncertain sequence. We establish a necessary and sufficient condition under which an infinite matrix transforms a convergent sequence into another. As an application of this notion, we prove the well-known Silverman–Toeplitz theorem and Kojima–Schur theorem in an uncertainty space.

The normal categories associated with unit regular semigroups and various classes of strongly unit regular semigroups were described as ${𝒰}{ℛ}$-categories with appropriate isomorphism extension properties. The normal categories associated with unit regular semigroups have been characterized as weak ${𝒰}{ℛ}$-categories. The cases of ${ℛ}$-strongly unit regular, ${ℒ}$-strongly unit regular and strongly unit regular semigroups were characterized using corresponding forms of isomorphism extensions. It is shown that when a normal category satisfies one of these isomorphism extension properties, then its normal dual satisfies a corresponding isomorphism extension property. Further, we give a description of the extension isomorphisms in the dual $N^{*}{𝒞}$ in terms of extensions in ${𝒞}$.

In this paper, we establish the orbit-stabilizer theorem and the orbit decomposition theorem for a Clifford semigroup. Moreover, we establish that for a Clifford semigroup $S$, any two homogeneous Clifford $S$-sets $X$ and $Y$ are Clifford $S$-isomorphic if and only if for any $x\in X,y\in Y$, $S_x$ is conjugate to $S_y$. More generally, for a Clifford semigroup $S$, any two Clifford $S$-sets $X$ and $Y$ are Clifford $S$-isomorphic if and only if there is a bijection $\beta :X/S\to Y/S$ satisfying the property that, for all $x\in X$, $[x]$ is Clifford $S$-isomorphic to $\beta ([x])$. Finally, we prove that in a Clifford semigroup $S$, for any element $a\in S$ such that $C(a)$ is $E$-unitary full Clifford subsemigroup of $S$, $[S:C(a)]\le |\mathrm{\text{cl}}(a)|$, where $\mathrm{\text{cl}}(a)$ is the conjugacy class in $S$ containing the element $a$.

In this paper, we utilize an inequality involving a coupled multivalued mapping and a singlevalued mapping to obtain a coupled coincidence point theorem. We discuss special conditions under which coupled common fixed point theorems are obtained. The result combines together several ideas prevailing in fixed point theory related studies. There are several corollaries and an illustrative example. In the second part of the paper, we establish a multivalued coupled coincidence point result by putting together several ideas. A contraction through an implicit relation for the coupled mapping is assumed in the main theorem for quadruples of points determined by certain functions. The coupled mapping is also required to satisfy certain dominated conditions which are defined in this paper. In Secs. 3 and 4, we study the data dependence and stability of coupled coincidence point sets, respectively. For describing set distance we use the Hausdorff metric. The work is in the domain of setvalued analysis.

In this paper, we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid $\mathrm{\text{wEnd}}{\overrightarrow{P}}_n$ of all weak endomorphisms of a directed path with $n$ vertices, which is a submonoid of the widely studied monoid ${{𝒪}}_n$ of all order-preserving transformations of an $n$-chain. Also, we describe the regular elements of $\mathrm{\text{wEnd}}{\overrightarrow{P}}_n$ and calculate its size and number of idempotents.

We extend integral transforms of Stieltjes type to a larger space of Boehmians than that used in previous studies. Our analysis includes the Stieltjes transform, the Poisson transform (Widder’s potential transform) and the generalized Stieltjes transform.

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph $G,$ are denoted by $D(G),D_Q(G)=D(G)+\mathrm{\text{Tr}}(G)$ and $D_L(G)=\mathrm{\text{Tr}}(G)-D(G),$ respectively, where $\mathrm{\text{Tr}}(G)$ is the diagonal matrix of vertex transmission. Ger$\check{\mathrm{\text{s}}}$gorin discs for any $n\times n$ square matrix $A=[a_{ij}]$ are the discs $\{z\in ℂ:|z-a_{ii}|\le R_i(A)\}$, where $R_i(A)={\sum }_{j\ne i}\left|a_{ij}\right|,i=1,2,\dots ,n$. The famous Ger$\check{\mathrm{\text{s}}}$gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger$\check{\mathrm{\text{s}}}$gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].

According to Lidstone’s interpolation theory, an entire function of a single variable of exponential type $<\pi$ is determined by it derivatives of even order at $0$ and $1$. In a previous paper, we gave a survey of this classical univariate theory. Here, we generalize it to two variables. Multivariate Lidstone interpolation will be the topic of a forthcoming paper.

Let $R$ be a prime or semiprime ring, $\lambda$ a nonzero left-sided ideal of $R$ and $\alpha ,\beta$ be two automorphisms of $R$. Let $F:R\to R$ and $G:R\to R$ be two generalized $(\alpha ,\beta )$-derivations of $R$ associated with $(\alpha ,\beta )$-derivations $d:R\to R$ and $g:R\to R$, respectively. The purpose of this paper is to investigate the following identities:

• (1)$G(xy)+d(x)F(y)+\alpha (xy)=0$;
• (2)$G(xy)+d(x)F(y)=0$;
• (3)$G(xy)+d(x)F(y)+\alpha (yx)=0$;
• (4)$G(xy)+d(x)F(y)+\alpha (yx)+\alpha (xy)=0$;
• (5)$G(xy)+d(x)F(y)+\alpha (yx)-\alpha (xy)=0$;
• (6)$G(xy)+d(y)F(x)+\alpha (yx)=0$;
• (7)$G(xy)+d(y)F(x)+\alpha (yx)+\alpha (xy)=0$;
• (8)$G(xy)+d(y)F(x)+\alpha (yx)-\alpha (xy)=0$;
• (9)$G(xy)+F(x)F(y)=0$;
• (10)$G(xy)+F(y)F(x)=0$;
• (11)$F(xy)+G(x)\alpha (y)+\alpha (yx)=0$;
• (12)$F(x)F(y)+G(x)\alpha (y)+\alpha (yx)=0$,

for all $x,y\in \lambda$.