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This paper deals with a class of singular problems involving the fractional $$p(x,\cdot )$$ -Laplace operator of the form $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p(x,\cdot )}u(x)= \frac{\lambda }{u^{\gamma (x)}}+u^{q(x)-1} &{} \hbox {in }\Omega , \\ u>0, \;\;\text {in}\;\; \Omega &{} \hbox {} \\ u=0 \;\;\text {on}\;\;{\mathbb {R}}^N\setminus \Omega , &{} \hbox {} \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $${\mathbb {R}}^N$$ ( $$N\ge 3$$ ), $$00$$ small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional $$p(x,\cdot )$$ -Laplace operators.

A graph Γ is called a bi-circulant if it admits a cyclic group as a group of automorphisms acting semiregularly on the vertices of Γ with two orbits. The characterization of tetravalent edgetransitive bi-circulants was given in several recent papers. In this paper, a classification is given of connected tetravalent vertex-transitive bi-circulants of order twice an odd integer.

In this paper, we will establish a formula that relates the product $$\prod _{\omega ^n=1} {\omega x-1 \atopwithdelims ()p-1}$$ to generalized and homogeneous multiple harmonic sums, this would allow us to derive new identities and congruences. The congruences considered in this paper are congruences in $$\mathbb {C}_p$$ which are also valid in $$\mathbb {Z}_p$$ and $$\mathbb {Z}$$ . In order to prove these congruences, we employ well-known theorems for symmetric functions and harmonic numbers.

Let λK m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K p,q -factorization of λK m,n is a set of edge-disjoint K p,q -factors of λK m,n which partition the set of edges of λK m,n . When p = 1 and q is a prime number, Wang, in his paper [On K 1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K 1,q -factorization of K m,n and gave a sufficient condition for such a factorization to exist. In papers [K 1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK m,n to have a K p,q -factorization. As a special case, it is shown that the necessary condition for the K p,q -factorization of λK m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.

By G(p, q) we denote a graph having p vertices and q edges, by V and E the vertex set and edge set of G respectively. A graph G(p, q) is said to have an edge magic labeling (valuation) with the constant (magic number) c(f) if there exists a one-to-one and onto function f: V ∪ E → {1, 2, …., p + q} such that f(u)+f(v)+f(uv) = c(f) for all uv ∈ E. An edge magic labeling f of G is called a super magic labeling if f(E) ={1, 2, …., q}. In this paper the concepts of the super magic and super magic strength of a graph are introduced. The super magic strength (sms) of a graph G is defined as the minimum of all constants c′(f) where the minimum is taken over all super magic labeling of G and is denoted by sms(G). This minimum is defined only if the graph has at least one such super magic labeling. In this paper, the super magic strength of some well known graphs P2n, P2n+1, K1,n, Bn,n, , Pn2 and (2n + 1)P2, Cn and Wn are obtained, where Pn is a path on n vertices, K1,n is a star graph on n+1 vertices, n-bistar Bn,n is the graph obtained from two copies of K1,n by joining the centres of two copies of K1,n by an edge e, if e is subdivided then Bn,n becomes , (2n + 1) P2 is 2n + 1 disjoint copies of P2, Pn2 is a square graph of Pn. Cn is a cycle on n vertices and Wn = Cn + K1 is wheel on n + 1 vertices.

Let $$\ell $$ be a fixed odd positive integer. In this paper, using some classical results on the generalized Ramanujan-Nagell equation, we completely derive all solutions (p, x, m, n) of the equation $$x^2=4p^n-4p^m+\ell ^2$$ with $$\ell ^21$$ , where p is a prime, x, m, n are positive integers satisfying $$\gcd (x,\ell )=1$$ and $$m4p^m$$ . As an example of using this method, we find all solutions (p, x, m, n) of the equation for $$\ell \in \{5,7\}$$ .

Let $$[\, \cdot \,]$$ be the floor function. In this paper we show that whenever $$\eta $$ is real, the constants $$\lambda _i$$ satisfy some necessary conditions, then for any fixed $$1

This paper considers comparison problems of predictor and estimator in the context of seemingly unrelated regression models ( $$\mathrm{SURM}$$ s). $$\mathrm{SURM}$$ s are a class of multiple regression equations with correlated errors among the equations from linear regression models. Our aim is to establish a variety of equalities and inequalities for comparing covariance matrices of the best linear unbiased predictors ( $$\mathrm{BLUP}$$ s) and the ordinary least squares predictors ( $$\mathrm{OLSP}$$ s) of unknown vectors under $$\mathrm{SURM}$$ s by using various rank and inertia formulas of block matrices. The results for comparisons of the best linear unbiased estimators ( $$\mathrm{BLUE}$$ s) and the ordinary least squares estimators ( $$\mathrm{OLSE}$$ s) in the models are also considered.

The goal of this paper is to describe the relationship between generalized B-opers, generalized $$\mathrm {SO}(2n,{\mathbb {C}})$$ -opers and (G, P)-opers. In particular, we show that to each generalized B-oper there is a naturally associated (G, P)-oper, but there are some (G, P)-opers that do not arise as generalized B-opers or $$\mathrm {SO}(2n,{\mathbb {C}})$$ -opers.

Let $$G_n:=U(n)\ltimes {\mathbb {H}}_n$$ be the semidirect product of the unitary group acting by automorphisms on the Heisenberg group $${\mathbb {H}}_n.$$ It was pointed out in [5], that the unitary dual $$\widehat{G_n}$$ of $$G_n$$ is homeomorphic to the space of admissible coadjoint orbits $${\mathfrak {g}}_n^\ddagger /G_n$$ of $$G_n.$$ One of the important subset of $$\widehat{G_n}$$ is what is called the cortex of $$G_n,$$ cor $$(G_n),$$ which is the set of all $$\pi \in \widehat{G_n}$$ that cannot be Hausdorff separated from the identity representation $$1_{G_n}$$ of $$G_n.$$ In the present paper, we use the orbit method of Lipsman to determine the cortex of $$G_n.$$

In this paper, criteria for strict convexity of Orlicz sequence spaces equipped with the p-Amemiya norms $$(1\le p\le \infty )$$ are presented. The obtained results widen the characterization of strict convexity of Orlicz sequence spaces. From our results it follows that there are Orlicz sequence spaces which are strictly convex for p-Amemiya norms with $$1

Slater introduced the point-addition operation on graphs to characterize 4-connected graphs. The $$\varGamma $$ -extension operation on binary matroids is a generalization of the point-addition operation. In this paper, for a given binary matroid M we characterize the closure operator and the flats of the $$\varGamma $$ -extension binary matroid $$M^X$$ in terms of the closure operator of original binary matroid M.

The edge-connectivity of a connected hypergraph H is the minimum number of edges (named as edge-cut) whose removal makes H disconnected. It is known that the edge-connectivity of a hypergraph is bounded above by its minimum degree. H is super edge-connected, if every edge-cut consists of edges incident with a vertex of minimum degree. A hypergraph H is linear if any two edges of H share at most one vertex. We call H uniform if all edges of H have the same cardinality. Sufficient conditions for equality of edge-connectivity and minimum degree of graphs and super edge-connected graphs are known. In this paper, we present a generalization of some of these sufficient conditions to linear and/or uniform hypergraphs.

A finite group $${{{\mathbb {G}}}}$$ is said to have perfect order subsets if for every d, the number of elements of $$ {{{\mathbb {G}}}}$$ of order d (if there are any) divides $$|{{{\mathbb {G}}}}|.$$ In this paper, we make a modest improvement on the result of Ford, Konyagin and Luca [3] about perfect order subsets.

In this paper, we consider compositions of integers when only parts of size at most 3 are allowed in both composition and its conjugate composition. First we obtain a relation between the number of such compositions and the Fibonacci numbers. Then we provide combinatorial identities between these compositions and compositions into 1’s and 2’s, compositions into odd parts, and compositions into parts greater than 1. Further, several generalized results for these compositions are obtained.

Let p be an odd prime, and let m be a positive integer satisfying $$p^m \equiv 3~(\text {mod }4).$$ Let $$\mathbb {F}_{p^m}$$ be the finite field with $$p^m$$ elements, and let $$R=\mathbb {F}_{p^m}[u]/\left\langle u^2\right\rangle$$ be the finite commutative chain ring with unity. In this paper, we determine all constacyclic codes of length $$4p^s$$ over R and their dual codes, where s is a positive integer. We also determine their sizes and list some isodual constacyclic codes of length $$4p^s$$ over R.

In this paper we characterize the algebraic structure of two-dimensional $$(\alpha ,\beta )$$ -constacyclic codes of arbitrary length $$s\ell $$ and of their duals over a finite field $$\mathbb{F}_q $$ , where $$\alpha ,\beta$$ are non zero elements of $$\mathbb{F}_q $$ . For $$\alpha ,\beta \in \{1,-1\}$$ , we give necessary and sufficient conditions for a two-dimensional $$(\alpha ,\beta )$$ -constacyclic code to be self-dual. We also show that a two-dimensional $$(\alpha ,1 )$$ -constacyclic code $${\mathcal {C}}$$ of length $$n=s\ell $$ cannot be self-dual if $$\gcd (s,q)= 1$$ . Finally, we give some examples of self-dual, isodual, MDS and quasi-twisted codes corresponding to two-dimensional $$(\alpha ,\beta )$$ -constacyclic codes.

Let P be a poset and $$\alpha :P\rightarrow P$$ be a function. The aim of this paper is to introduce and study the notion of skew derivations on P. We prove some fundamental properties of posets involving skew derivations. In particular, apart from proving the other results, we prove that if d and g are two skew derivations of P associated with an automorphism $$\alpha $$ such that $$d\alpha =\alpha d$$ and $$g\alpha =\alpha g,$$ then $$d \le g $$ if and only if $$g d =\alpha d$$ . Also, we prove that $$ Fix_{\alpha ,d}(P)\cap l(\alpha (x)) = l(d(x))$$ for all $$x\in P.$$ Furthermore, we give some examples to demonstrate that various restrictions imposed in the hypotheses of our results are not superfluous.

In this paper, we establish a non-trivial upper bound for the exponential sum $$\begin{aligned} \sum _{y

In this paper, we define a graph operation, namely, M-edge rooted product of graphs. This generalizes the existing graph operation called graphs with edge pockets. Also we introduce a matrix invariant, namely, coronal of a matrix constrained by the index sets. We compute this value for some class of matrices with respect to some index sets. We obtain the generalized characteristic polynomial of the graph obtained by M-edge rooted product with a help of this invariant. Consequently, we deduce the characteristic polynomial of the adjacency matrix, the Laplacian matrix and the signless Laplacian matrix of this graph. Using these results, we derive the L-spectrum of several families of M-edge rooted product of graphs and deduce several existing results on the spectra of graphs with edge pockets in the literature. As applications, we obtain infinitely many L-cospectral graphs and construct A-integral graphs, L-integral graphs.

Mennicke–Newman lemma for unimodular rows was used by W. van der Kallen to give a group structure on the orbit set $$\frac{Um_{n}(R)}{E_{n}(R)}$$ for a commutative noetherian ring of dimension $$d\le 2n-4.$$ In this paper, we generalise the Mennicke–Newman lemma for $$m\times n $$ right invertible matrices

In this paper, we consider the class of polynomials $$P(z):=\sum \limits _{j=0}^{n}c_jz^j$$ having all zeros in a closed disk $$|z|\le k,\text {where}~ k\ge 1$$ and obtain a result that improves and generalizes the results of Govil, Jain and others by using certain coefficients of P(z).

Let A be a nonempty finite set of integers. For a real number m, the set $$m\cdot A=\{ma: a\in A\}$$ denotes the set of m-dilates of A. In 2008, Bukh initiated an interesting problem of finding a lower bound for the sumset of dilated sets, i.e., a lower bound for $$|m_1\cdot A+m_2\cdot A+ \cdots +m_n\cdot A|$$ , where $$m_1, m_2, \ldots , m_n$$ are integers. In this paper, we consider the case $$|3\cdot A+k\cdot A|$$ , where k is a prime number $$\ge 5$$ . Under some assumptions on A, first we give a general lower bound on the cardinality of $$3\cdot A+k\cdot A$$ then, for large sets A, under the same assumptions, we improve this general lower bound. The results also hold true for the general sumset $$q \cdot A + k\cdot A$$ , where q is any odd prime $$< k$$ , under the similar assumptions on A.

In this paper, we prove the strong log-concavity and the unimodality of the various sequences of an extension of elementary symmetric function. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot. As applications, we establish the strong q-log-concavity and the unimodality of q-bi $$^{s}$$ nomial coefficients and their sequences lying on rays of the associated triangle.

There is no known formula that counts the number of transitive relations on a set with n elements. In this paper, it is shown that no polynomial in n with integer coefficients can represent a formula for the number of transitive relations on a set with n elements. Several inequalities giving various useful recursions and lower bounds on the number of transitive relations on a set are also proved.

The adjacency matrix of a simple and undirected graph G is denoted by $${\mathcal {A}}(G)$$ and $${\mathcal {D}}_{G}$$ is the degree diagonal matrix of G. The Laplacian matrix of G is $${\mathcal {L}}(G)={\mathcal {D}}_{G}-{\mathcal {A}}(G)$$ and the signless Laplacian matrix of G is $${\mathcal {Q}}(G)={\mathcal {D}}_{G}+{\mathcal {A}}(G) $$ . The star graph of order n is denoted by $$S_{n}$$ . The double starlike tree $${\mathcal {G}}_{p,n,q}$$ is obtained by attaching p pendant vertices to one pendant vertex of the path $$P_n$$ and q pendant vertices to the other pendant vertex of $$P_n$$ . In this paper, we first investigate the disjoint union of double starlike graphs $${\mathcal {G}}_{p,2,q}$$ and the star graphs $$S_{n}$$ for Laplacian (signless) spectral characterization. Also, the signless Laplacian spectral determination of the disjoint union of odd unicyclic graphs and star graphs is studied. Abdian et al. [AKCE Int. J. Graphs Combin. (2018) https://doi.org/10.1016/j.akcej.2018.06.009] proved that if G is a $$\mathcal {DQS}$$ connected non-bipartite graph with $$n\ge 3$$ vertices, then $$G\cup rK_{1}\cup sK_{2}$$ is $$\mathcal {DQS}$$ . Here we give a counterexample for the claim and also we study the graph $$G\cup rK_{1}\cup sK_{2}$$ for signless Laplacian charcterization when G has at least $$((n-2)(n-3)+10)/2$$ edges and $$s=1$$ . It is shown that the graph $$K_{n}\cup K_{2}\cup rK_{1}$$ is $$\mathcal {DQS}$$ for $$n\ge 4$$ . We also prove that the complement graph of $$K_{n}\cup K_{2}\cup rK_{1}$$ is $$\mathcal {DQS}$$ for $$r>1$$ and $$n\ne 3$$ .

A graph H of order n is said to be $$k-placeable$$ into a graph G, having the same order n, if G contains k edge-disjoint copies of H. Kaneko et al. [9] proved that any non-star tree T is $$2-placeable$$ into its third power $$T^3$$ . In this paper, we give a particular interest on the $$3-placement$$ of a tree T into its sixth power $$T^6$$ .

In this paper, one-parameter families $${\mathcal {F}}\equiv \left\{ f_{\lambda }(z)=\lambda \left( \cosh z+\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\} $$ and $${\mathcal {G}}\equiv \left\{ g_{\lambda }(z)=\lambda \left( \cosh z-\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\} $$ are considered and the dynamics of functions $$f_{\lambda }\in {\mathcal {F}}$$ and $$g_{\lambda }\in {\mathcal {G}}$$ are investigated. It is shown that both the functions $$f_{\lambda }$$ and $$g_{\lambda }$$ have finite number of singular values and the origin is always an attracting fixed point of $$g_{\lambda }(z)$$ . The dynamics of $$f_{\lambda }(z)$$ and $$g_{\lambda }(z)$$ on the extended complex plane are studied by investigating the nature of the real fixed points and the singular values of $$f_{\lambda }$$ and $$g_{\lambda }$$ . It is shown that a bifurcation and chaotic burst occur at a certain parameter value of $$\lambda $$ for the functions $$f_{\lambda }$$ in the family $${\mathcal {F}}$$ but there is no bifurcation in the family $${\mathcal {G}}$$ .

In this paper, we are interested in the existence and multiplicity of positive solutions for the following Kirchhoff–Schr $$\ddot{\text {o}}$$ dinger–Newton system $$\begin{aligned} {\left\{ \begin{array}{ll} -\left( 1+b\displaystyle \int _\Omega |\nabla u|^2dx\right) \Delta u=(\lambda _1+\delta )u+\phi |u|u, &{} \text {in}\ \ \Omega , \\ -\Delta \phi =|u|^3,&{} \text {in}\ \ \Omega , \\ u=\phi =0, &{} \text {on}\ \ \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^{4}$$ is a smooth bounded domain, $$b>0$$ , $$\delta >0$$ , $$\lambda _1$$ is the first eigenvalue of $$-\Delta$$ on $$\Omega$$ , and two positive solutions are established via variational method.

Let $$G_n$$ and $$G'_{n}$$ be two nonisomorphic graphs on n vertices with spectra $$\begin{aligned} \lambda _{1} \geqslant \lambda _{2} \geqslant \cdots \geqslant \lambda _{n} \ \ \mathrm {and} \ \ \lambda '_{1} \geqslant \lambda '_{2} \geqslant \cdots \geqslant \lambda '_{n}, \end{aligned}$$ respectively. Define the distance between the spectra of $$G_{n}$$ and $$G'_{n}$$ as $$\begin{aligned} \lambda (G_{n}, G'_{n})= \sum \limits _{i = 1}^n (\lambda _{i}- \lambda _{i}')^{2} \ \mathrm { \big( or \ use \ } \sum \limits _{i = 1}^n \vert \lambda _{i}- \lambda _{i}' \vert \big). \end{aligned}$$ Let $$\begin{aligned} \mathrm {cs}(G_{n})= \mathrm {min} \{ \lambda (G_n, G'_{n}): G'_{n} \ \mathrm { not \ isomorphic \ to} \ G_{n} \}, \end{aligned}$$ and $$\begin{aligned} \text{cs}_{n}=\max \{\text{cs}(G_{n}): G_{n} \text{ a graph on } n \text{ vertices}\}. \end{aligned}$$ Richard Brualdi in [9] proposed the following problems: Problem A. Investigate cs( $$G_{n}$$ ) for special classes of graphs. Problem B. Find a good upper bound on $$\mathrm {cs}_n$$ . In this paper, we investigate problem A and determine cs( $$G_{n}$$ ), when $$G_n$$ is a graph of order n with size two or three. Let $$K_{n}+ K_{1}$$ be a disjoint union of the complete graph $$K_n$$ with one isolated vertex, and $$K_{n+1}^{1}$$ be a complete graph $$K_{n+1}$$ by deleting $$n-1$$ edges from one vertex of $$K_{n+1}$$ . We show that if $$\lambda (K_{n}+ K_{1}, G_{n}')=$$ cs( $$G_{n}$$ ), then $$G_{n}'$$ is isomorphic to $$K_{n+1}^{1}$$ .

Let $$R_1(n), R_2(n)$$ denote the numbers of representations of a positive integer n by the quaternary quadratic forms $$g_1(x_1,x_2,x_3,x_4)$$ = $$2( x_{1}^{2}+x_1 x_2+ x_{2}^{2})+2x_1x_3 +x_1x_4+ x_2x_3+2x_2x_4+2(x_{3}^{2}+x_3 x_4+x_{4}^{2}), g_2(x_{1},x_2,x_3,x_4)=8( x_{1}^{2}+x_{2}^{2})+x_{3}^{2}+x_{4}^{2}$$ , respectively, where $$x_1$$ , $$x_2$$ , $$x_3$$ and $$x_4$$ are integers. In this paper, we establish the asymptotic formulae for the sums $$\sum \limits _{n\le x}R_i(n)$$ and $$\sum \limits _{n\le x}R_i^{2}(n)$$ for $$i=1,2$$ .

The anti-Gallai graph $$\Delta (G)$$ of a graph G, has the edges of G as its vertices and two distinct edges are adjacent in $$\Delta (G)$$ if they lie on a triangle in G. In this paper we characterize the graphs whose anti-Gallai graph has only one or two eigenvalues and study the adjacency spectrum of the anti-Gallai graph of $$(K_4,H)$$ -free graphs, where H is the Hajos graph and graphs in which two diamonds share at most one vertex. We also find expressions for the adjacency spectrum of anti-Gallai graph of graphs after applying certain binary operations.

In this paper, we consider the title simultaneous Pell equations, where $$a\ge 2,b \ge 1$$ are positive integers and p is an odd prime. Particularly, we give all the solutions when $$2\le a\le 1000$$ and $$1\le b\le 1000$$ .

A simple undirected graph $$\Gamma _G$$ whose vertex set is the set of non-central elements of a finite group G and two vertices x and y are adjacent if they commute is called commuting graph of G. In this paper, we compute energy, Laplacian energy and signless Laplacian energy of $$\Gamma _G$$ for some classes of super integral groups.

In this paper, we study the following planar Schrodinger-Newton system with a Coulomb potential $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+ W(x)u+ 2\pi \phi u+\int _{{\mathbb {R}}^{2}}\frac{[u(y)]^{2}}{|x-y|}dyu = f(u), &{} \text{in}\,{\mathbb {R}}^{2},\\ \Delta \phi =u^2, &{} \text{in}\,{\mathbb {R}}^{2}, \end{array}\right. \end{aligned}$$ where f is super-linear at zero and subcritical at infinity. We obtain infinitely many sign-changing solutions of the above problem via the invariant sets of descending flow.

Let $$p_{\omega }(n)$$ be the number of partitions of n in which each odd part is less than twice the smallest part, and assume that $$p_{\omega }(0)=0$$ . Andrews, Dixit, and Yee proved that the generating function of $$p_{\omega }(n)$$ equals $$q\omega (q)$$ , where $$\omega (q)$$ is one of the third order mock theta functions. Many scholars have studied the arithmetic properties of $$p_{\omega }(n)$$ . For example, Waldherr proved that $$p_{\omega }(40n+28)\equiv p_{\omega }(40n+36)\equiv 0 \pmod 5$$ by using the theory of Maass forms, which was later proved once again in an elementary method by Andrews, Passary, Sellers, and Yee. In this paper, we shall establish four new congruences modulo 5 for $$p_{\omega }(n)$$ .

In this paper we characterize groups according to the number of end vertices in the associated coprime graphs. An upper bound on the order of the group that depends on the number of end vertices is obtained. We also prove that $$2-$$ groups are the only groups whose coprime graphs have odd number of end vertices. Classifications of groups with small number of end vertices in the coprime graphs are given. We give a complete answer to [4, Question 3.7], where we show that $$\mathbb {Z}_4$$ and $$\mathbb {Z}_2\times \mathbb {Z}_2$$ are the only groups whose coprime graph has exactly three end vertices.

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

A graph G is a prime distance graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is a prime number. It is known that cycles and bipartite graphs are prime distance graphs. In this paper we derive some general results concerning prime distance labeling of graphs and also establish interesting results for complete graphs, wheel graphs, and wheel-related graphs.

Let G be a finite graph, simple, undirected and has no isolated vertex. A dominating set D of G is said a pitchfork dominating set (pds) if every v in it dominates at most k and at least j vertices out of D, for any non-negative integers j and k. The pitchfork domination number (pdn) of G, denoted by $$\gamma _{pf}(G)$$ is the order of a minimum pds in G. In this paper, the inverse pitchfork domination is introduced, the definition is used for $$j=1$$ and $$k=2$$ . Several bounds are determined that are associated with size, order, maximum degree and minimum degree of a graph. Also, this type of domination on some known graphs is calculated, like: path, cycle, complete, complete bipartite and wheel graph. The essential conditions for any graph having an inverse pitchfork domination are discussed.

In this paper, we carry the obtained results for a certain class of Moufang-Klingenberg planes co-ordinatized by the local alternative ring of dual numbers, over to a class of Moufang-Klingenberg planes coordinatized by the plural algebra of order m which is a local alternative ring. So, we show that its collineation group is transitive on quadrangles and therefore that coordinatization of the planes is independent of the choice of the coordinatization quadrangle. Moreover, by a conjugacy class we discuss the definition of cross-ratio of four distinct points on the projective lines for the planes.

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. Let p be a prime. In this paper, we proved that there is only one connected heptavalent symmetric graphs of order 40p, it is a vertex primitive graph of order 40.3 = 120 admitting S7 as its full automorphism group.

In this paper we investigate the compact operators under Orlicz function, named noncommutative Orlicz sequence space (denoted by Sϕ(ℌ)), where ℌ is a complex, separable Hilbert space. We will show that the space generalizes the Schatten classes Sp(ℌ) and the classical Orlicz sequence space respectively. After getting some relations of trace and norm, we will give some operator inequalities, such as Holder inequality and some other classical operator inequalities. Also we will give the dual space and reflexivity of Sϕ(ℌ) which generalizes the results of Sϕ(ℌ). Finally, as an application, we will show that the Toeplitz operator $${T_{1 — {{\left| z \right|}^2}}}$$ on the Bergman space $$L_\alpha ^2\left(\mathbb{R} \right)$$ belongs to some Sϕ(ℌ), and the norm satisfies $$1 = \sum\limits_{n \ge 0} {\varphi \left({{1 \over {\left({n + 2} \right){{\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|}_\varphi}}}} \right)} $$ . Especially, if ϕ(T) = ∣T∣p, p > 1, the norm is $${\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|_p} = {\left[{\sum\limits_{n \ge 0} {{1 \over {{{\left({n + 2} \right)}^p}}}}} \right]^{{1 \over p}}} = {\left({\zeta \left(p \right) — 1} \right)^{{1 \over p}}}$$ , where ζ(p) is the Riemann function.

The Gallai graph Γ(G) of a graph G, has the edges of G as its vertices and two distinct vertices are adjacent in Γ(G) if they are adjacent edges in G, but do not lie on a triangle. In this paper we find the adjacency spectrum of Gallai graph of some graphs derived from simple graphs by certain operations, the characteristic polynomial for some graphs and exhibit the adjacency matrix of some other graphs.

In this paper, we prove that if R is a commutative ring with unity and M is a finitely generated R-module, then M is Noetherian if and only if for every prime ideal P of R with $$Ann(M) \subseteq P$$ , there exists a finitely generated submodule $$N_P$$ of M such that $$PM \subseteq N_P \subseteq M(P)$$ .

Let $$\mathfrak{F}$$ denote a class of groups. A maximal subgroup M of G is called $$\mathfrak{F}$$ -abnormal provided G/MG ∉ $$\mathfrak{F}$$ . We say that (K, H) is an $$\mathfrak{F}$$ -abnormal pair of G provided K is a maximal $$\mathfrak{F}$$ -abnormal subgroup of H. Let Σ = {G0 ≤ G1 ≤ G2 ≤ … ≤ Gn} be a subgroup series of G. A subgroup H of G is said to be Σ- $$\mathfrak{F}$$ -embedded in G if H either covers or avoids every $$\mathfrak{F}$$ -abnormal pair (K, H) such that Gi−1≤ K < H ≤ Gi for some i ∈ {0, 1, …, n}. In this paper, some new characterizations of p-supersoluble and p-soluble are given by discussing the properties of Σ- $$\mathfrak{F}$$ -embedded of subgroups.

In this paper, in considering aspects of the geometric mean sequence, offers new results connecting generalized Tribonacci and third-order Horadam numbers which are established and then proved independently.

Let $$H(p,tK_{1,m}^ * )$$ be a connected unicyclic graph with p + t(m + 1) vertices obtained from the cycle Cp and t copies of the star K1, m by joining the center of K1, m to each one of t consecutive vertices of the cycle Cp through an edge, respectively. When t = p, the graph is called a dandelion graph and when t ≠ p, the graph is called a broken dandelion graph. In this paper, we prove that the dandelion graph $$H(p,pK_{1,m}^ * )$$ and the broken dandelion graph $$H(p,tK_{1,m}^ * )$$ (0 < t < p) are determined by their Laplacian spectra when m ≠ 2 and p is even.

Let G be a connected graph with n vertices, m edges and having distance signless Laplacian eigenvalues ρ1≥ ρ2 ≥ … ≥ ρn≥ 0. For any real number α ≠ 0, let $${m_\alpha }\left( G \right) = \sum\nolimits_{i = 1}^n {\rho _i^\alpha } $$ be the sum of αth powers of the distance signless Laplacian eigenvalues of the graph G. In this paper, we obtain various bounds for the graph invariant mα(G), which connects it with different parameters associated to the structure of the graph G. We also obtain various bounds for the quantity DEL(G), the distance signless Laplacian-energy-like invariant of the graph G. These bounds improve some previously known bounds. We also pose some extremal problems about DEL(G).

In this paper, we are interested in considering the following singular elliptic problem with concaveconvex nonlinearities $$\left\{ {\begin{array}{*{20}{l}} { - \Delta u - \frac{\mu }{{|x{|^2}}}u = f(x)|u{|^{p - 2}}u + g(x)|u{|^{q - 2}}u,}&{in\;\;\Omega \backslash \{ 0\} ,} \\ {u = 0,}&{on\;\;2\Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN(N ≥ 3) is a smooth bounded domain with 0 ∈ Ω, $$0 < \mu < \bar\mu = \frac{{{{(N - 2)}^2}}}{4}$$, 1 < q < 2 < p < 2* and $$2* = \frac{{2N}}{{N - 2}}$$ is the Sobolev critical exponent, the coefficient functions f, g may change sign on Ω. By the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.