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It has been proven that there exists a one-to-one correspondence H(t, x) between solutions of the linear system and the nonlinear system in the previous work. However, there is no paper considering the Holder regularity of the transformation H(t, x) in the literature. This paper fills the gap. We establish a strict proof of the Holder regularity of the transformation H(t, x). We show that the conjugating function H(t, x) in the generalized Hartman–Grobman theorem is always Holder continuous.

In the present paper, we study linear operators \(\Delta \) from the algebra of \(2\times 2\) matrices \({\mathbb {M}}_2({\mathbb {C}})\) into its tensor square. Each such kind of mapping defines a quadratic operator on the state space of \({\mathbb {M}}_2({\mathbb {C}})\). We know that q-purity of quasi quantum quadratic operators (q.q.o.) is equivalent to the invariance of the unite sphere under the corresponding quadratic operator. Therefore, in the paper, we consider quadratic operators, which preserve the unit circle, and show that the corresponding quasi q.q.o. cannot be not positive. Note that this is a much weaker condition than the q-purity of quasi q.q.o. Moreover, we will classify q-pure circle preserving quadratic operators into three disjoint classes (non isomorphic). Moreover, we are able to show that quasi q.q.o. corresponding to the first class is block positive. Note that the block positivity is weaker than positivity. This kind of operator, i.e., not positive but block-positive operator allows us to detect that the given state on \({\mathbb {M}}_2({\mathbb {C}})\otimes {\mathbb {M}}_2({\mathbb {C}})\) is either entangled or not. The obtained results will allow us to verify whether a given mapping is positive or not. This finding suggests us to produce a class of non-positive mappings. Moreover, it will shed some light in finding entanglement states.

In this paper, we give some properties of the generalized derivation algebra \(\mathrm{GDer}(T)\) of a Hom–Lie triple systems T. In particular, we prove that \(\mathrm{GDer}(T) = \mathrm{QDer}(T) + \mathrm{QC}(T)\), the sum of the quasiderivation algebra and the quasicentroid. We also prove that \(\mathrm{QDer}(T)\) can be embedded as derivations in a larger Hom–Lie triple system. General results on centroids of Hom–Lie triple systems are also developed in this paper.

Consider a discrete-time supercritical branching random walk, which is a branching process combined with random spatial motion of particles, the number of the descendants tending to infinity with positive probability. Let $$ Z_n(\cdot ) $$ be the counting measure which counts the number of particles of generation n in a given set. Revesz (1994) studied the convergence rates in the central and local limit theorems for $$ Z_n(\cdot ) $$ in some special cases, and then, the topic was further developed in various cases. In this paper, we give the exact convergence rates of the central and local limit theorems for $$ Z_n(\cdot ) $$ in the case the underlying motion is governed by a general non-lattice random walk on $${\mathbb {R}}^d$$ with the characteristic function of the motion law satisfying the weak Cramer condition.

Linear complementary dual (LCD) codes are linear codes whose intersections with their duals are trivial. In this paper, characterizations of LCD codes with respect to the symplectic inner product, i.e. symplectic LCD codes, over finite fields are given. Some methods for constructing symplectic LCD codes and symplectic LCD MDS codes are presented. As an application, a class of symplectic LCD MDS codes is constructed by employing Vandermonde matrices, and the corresponding MDS maximal entanglement entanglement-assisted quantum error-correcting codes (EAQECCs) are constructed.

In this paper, we mainly discuss the constructions of some new K-g-frames which differ from the existing methods. Meanwhile, we use the relation between a positive operator and the frame operator of a K-g-frame to yield a new K-g-frame. We also obtain a necessary and sufficient condition to generate a new K-g-frame. In addition, we correct some recent results which were obtained by Huang and Leng. In the end, we give an equivalent characterization to construct some new tight K-g-frames by two given g-Bessel sequences. Our results generalize and improve some remarkable results.

In this paper we give an estimate for the difference between the moduli of two roots of a polynomial with integer coefficients in terms of its degree and Mahler measure. An application of this estimate implies a stronger version of a recent result of Gomez Ruiz and Luca. In passing, we prove an estimate for the distance between two algebraic numbers in terms of their degrees and Mahler measures. For possible applications all our results are given with explicit constants. They are stated without any extra conditions or unnecessary assumptions whenever possible.

In this paper, we prove weak and strong convergence theorems of SP-iteration for common fixed point of three G-nonexpansive mappings in uniformly convex Banach spaces endowed with a directed graph under some suitable control conditions. We also give some numerical examples for confirming our main theorem and compare convergence rate between SP-iteration and Noor iteration.

In this paper, we concern with the following Schrodinger–Poisson system: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\phi u = f(x,u)+u^5 ,\quad &{} x\in \Omega ,\\ -\Delta \phi =u^2,\quad &{} x\in \Omega ,\\ u=\phi =0, \quad &{} x \in \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $$\mathbb {R}^{3}$$ and $$u^5$$ reaches the Sobolev critical exponent since $$2^*=6$$ in dimension 3. Under some appropriate assumptions on f, a new result on the existence of nontrivial solutions is obtained via variational methods.

The Laplacian spread of a graph G is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of G. Let $${\mathcal {Q}}_t(n, d)=\{G\, |\, G-v_0\ \hbox {is a tree on }n-1\hbox { vertices and } d_G(v_0)=d\}.$$ Recently, Y. Xu and J. Meng characterized the unique graph with maximum Laplacian spread among all graphs in the set $${\mathcal {Q}}_t(n, d)$$ with $$1\le d\le (n-4)/2$$ . In this paper, we extend their result by determining the unique graph with maximum Laplacian spread among all graphs in the set $${\mathcal {Q}}_t(n, d)$$ with $$1\le d\le n-5$$ .

Let G be a simple connected graph. The eccentric complexity of graph G is introduced as the number of different eccentricities of its vertices. A graph with eccentric complexity equal to one is called self-centered. In this paper, we study the eccentric complexity of graph under several graph operations such as complement of graph, line graph, Cartesian product, sum, disjunction and symmetric difference of graphs. Also, we present an infinite family of non-vertex-transitive self-centered graphs and we prove that all such graphs are 2-connected. Further, for any D and k where $$k \le \frac{D-1}{2}$$ , we construct an infinite family of non-vertex-transitive graphs with eccentric complexity k and diameter D. Extremal graphs with minimum or maximum total eccentricity among all graphs with given eccentric complexity are determined. We also consider a family of nanotubes and show that it is extremal with respect to the eccentric complexity among all fullerene graphs. At the end we also indicate some possible directions of further research.

In this paper, we consider the following semilinear elliptic systems: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u=F_{u}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$ where $$V:\mathbb {R}^{N}\rightarrow \mathbb {R},~F_{u}(x,u,v)$$ and $$F_{v}(x,u,v)$$ are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $$-\triangle +V$$ . Under appropriate assumptions on $$F_{u}(x, u, v)$$ and $$F_{v}(x, u, v)$$ , we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended.

Let $$(P,\le )$$ be a partially ordered set with a least element 0. The regular digraph of ideals of P, denoted by $$\overrightarrow{\Gamma }(P)$$ , is a digraph whose vertex-set is the set of all nontrivial ideals of P and, for every two distinct vertices I and J, there is an arc from I to J whenever I contains a nonzero zero-divisor on J. Also, the undirected regular graph of ideals of P, denoted by $$\Gamma (P)$$ , is a graph with all nontrivial ideals of P being the vertex-set, and for two distinct vertices I and J, I is adjacent to J if and only if I contains a nonzero zero-divisor on J or J contains a nonzero zero-divisor on I. In this paper, we study some basic properties of $$\overrightarrow{\Gamma }(P)$$ . Also we completely characterize all posets P with planar regular graph.

Classical topological indices, such as Zagreb indices ( $$M_{1}$$ and $$M_2$$ ) and the well-studied eccentric connectivity index ( $$\xi ^{c}$$ ) directly or indirectly consider the total contribution of all edges in a graph. By considering the total degree sum of all non-adjacent vertex pairs in a graph, Ashrafi et al. (Discrete Appl Math 158:1571–1578, 2010) proposed two new Zagreb-type indices, namely the first Zagreb coindex ( $$\overline{M}_{1}$$ ) and second Zagreb coindex ( $$\overline{M}_{2}$$ ), respectively. Motivated by Ashrafi et al., we consider the total eccentricity sum of all non-adjacent vertex pairs, which we call the eccentric connectivity coindex ( $$\overline{\xi }^{c}$$ ), of a connected graph. In this paper, we study the extremal problems of $$\overline{\xi }^{c}$$ for connected graphs of given order, connected graphs of given order and size, and the trees, unicyclic graphs, bipartite graphs containing cycles and triangle-free graphs of given order, respectively. Additionally, we establish various lower bounds for $$\overline{\xi }^{c}$$ in terms of several other graph parameters.

For $$n\in \mathbb {N}$$ , let $$O_{n}$$ be the semigroup of all order-preserving transformations on the finite chain $$X_{n}=\{1,\ldots ,n\}$$ , under its natural order. For any non-empty subset A of $$X_{n}$$ , let $$O_{n}(A)$$ and $$O_{n}^+(A)$$ be the subsemigroups of all order-preserving and A-decreasing, and of all order-preserving and A-increasing transformations on $$X_{n}$$ , respectively. In this paper we obtain formulae for the number of elements and for the number of idempotents in $$O_{n}(A)$$ . Moreover, we show that $$O_{n}(A)$$ contains a zero element if and only if $$1\in A$$ , and then we obtain the number of nilpotents in $$O_{n}(A)$$ when $$1\in A$$ .

Let R be a commutative ring. The Gorenstein super finitely presented dimension of R is defined as $$G\hbox {-}\text{ s.gl. }\dim (R)=\sup \{\text{ Gpd }_{R}{M}\,|\,M$$ is a super finitely presented R-module}. In this paper, we give a series of characterizations of the ring with $$G\text{- }\text{ s.gl }.\dim (R)=0$$ . We show that R is a ring with $$G\text{- }\text{ s.gl }.\dim (R)=0$$ if and only if R is self-weak injective and R is a generalized coherent ring (i.e., R is a GFC-ring), where an R-module M is called weak injective if $$\mathrm{Ext}_{R}^{1}(N,M)=0$$ for every super finitely presented R-module N, and R is called a generalized coherent ring if $$M^{*}$$ is super finitely presented for any super finitely presented R-module M. Further, comparing with FGF-question, we give an example to show that a ring R is not necessarily a QF-ring if every (super) finitely presented R-module can embedded in a free R-module.

It was previously shown that there is a bijection between the family of locally disconnected 2-geodesic transitive graphs $$\Gamma $$ and a certain family of partial linear spaces $${\mathcal {S}}(\Gamma )$$ . In this paper, we first determine a relationship between the 2-geodesic transitivity of $$\Gamma $$ and the local s-arc transitivity of the incidence graph of $${\mathcal {S}}(\Gamma )$$ . Next, we give a reduction theorem for the family of locally disconnected s-geodesic transitive graphs.

In this paper, we studied a normality criterion concerning Hayman’s question and proved: let $$n({\ge }2),k({\ge }1), m({\ge }0)$$ be three integers, let $$h(z)({\not \equiv } 0)$$ be a holomorphic function in a domain D with all zeros that have multiplicity at most m, and let $$\mathcal {F}$$ be a family of functions meromorphic in a domain D, all of whose zeros have multiplicity at least $$k+m$$ . If, for any two functions $$f, g\in \mathcal {F}$$ , $$f^nf^{(k)}$$ and $$g^ng^{(k)}$$ share h(z) in D, then $$\mathcal {F}$$ is normal in D. The result gets rid of two conditions “all zeros of h(z) have multiplicity divisible by $$n+1$$ ” and “all poles of f(z) have multiplicity at least $$m+1$$ ” in the result due to Meng and Hu (Bull Malays Math Sci Soc 38:1331–1347, 2015).

In this paper, we develop sufficient criteria for the existence of a nonoscillatory solution to the fractional neutral functional differential equation of the form: $$\begin{aligned} D^{\alpha }_t[x(t)+c x(t-\tau )]'+\sum ^m_{i=1}P_i(t)F_i(x(t-\sigma _i))=0,\quad t\ge t_0, \end{aligned}$$ where \(D_t^{\alpha }\) is Liouville fractional derivatives of order \(\alpha \ge 0\) on the half-axis, \(c\in \mathbb {R}\), \(\tau \), \(\sigma _i\in \mathbb {R}^+\), \(P_i\in C([t_0, \infty ), \mathbb {R})\), \(F_i\in C(\mathbb {R}, \mathbb {R}), ~ i=1,2,\ldots ,m\), \(m \ge 1\) is an integer. Our results are new and improve many known results on the integer-order functional differential equations.

In the present paper, we introduce a two parametric q-analogue of Stancu-Beta operators and establish some direct results in the polynomial weighted space of continuous functions defined on the interval \([0,\infty )\). We use Lipschitz-type maximal function to find pointwise estimate. Furthermore, we obtain a Voronovskaja-type theorem for these operators.

In this paper, we give some new essential norm estimates of weighted composition operators \(uC_{\varphi }\) from analytic Besov spaces into the Bloch space, where u is a function analytic on the unit disk \(\mathbb {D}\) and \(\varphi \) is an analytic self-map of \(\mathbb {D}\). Moreover, new characterizations for the boundedness, compactness and essential norm of weighted composition operators \(uC_{\varphi }\) are obtained by the nth power of the symbol \(\varphi \) and the Volterra operators \(I_u\) and \(J_u\).

In this paper, we study the value distribution of differential polynomial with the form \(f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)},\) where f is a transcendental meromorphic function. Namely, we prove that \(f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)}-P(z)\) has infinitely zeros, where P(z) is a nonconstant polynomial and \(n\in {\mathbb {N}},\)\(k, n_1, \dots , n_k, t_1, \dots , t_k\) are positive integer numbers satisfying \(n+\sum _{v}^{k}n_v\ge \sum _{v=1}^{k}t_v+3.\) Using it, we establish some normality criterias for family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. Our results generalize some previous results on normal family of meromorphic functions.

A proper edge coloring is called acyclic if no bichromatic cycles are produced. It was conjectured that every simple graph G with maximum degree $$\varDelta $$ is acyclically edge- $$(\varDelta +2)$$ -colorable. Basavaraju and Chandran (J Graph Theory 61:192–209, 2009) confirmed the conjecture for non-regular graphs G with $$\varDelta =4$$ . In this paper, we extend this result by showing that every 4-regular graph G without 3-cycles is acyclically edge-6-colorable.

A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors that are needed to make G proper connected. In this paper, we study the proper connection number on the lexicographic, strong, Cartesian, and direct products and present exact values or upper bounds for these products of graphs.

In this paper, we investigate the properties of right orthogonal modules of $${{\mathscr {C}}}$$ , where $${{\mathscr {C}}}$$ is a class of left R-modules. As an application, we investigate the properties of right orthogonal modules of Ding injective left R-modules, and present various characterizations of semisimple and von Neumann regular rings and so on. Moreover, we also consider another cohomology, strong Tate cohomology, which connects the usual cohomology with the Ding cohomology.

Let R be a commutative ring with identity and $$\hbox {Spec}^{s}(M)$$ denote the set all second submodules of an R-module M. In this paper, we investigate various properties of $$\hbox {Spec}^{s}(M)$$ with respect to different topologies. We investigate the dual Zariski topology from the point of view of separation axioms, spectral spaces and combinatorial dimension. We establish conditions for $$\hbox {Spec}^{s}(M)$$ to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop.

In this paper we identify the structure of complex finite-dimensional Leibniz algebras with associated Lie algebras $$sl_2^1\oplus sl_2^2\oplus \dots \oplus sl_2^s\oplus R,$$ where R is a solvable radical. The classifications of such Leibniz algebras in the cases $$\mathrm{dim} R=2, 3$$ and $$\mathrm{dim} I\ne 3$$ have been obtained. Moreover, we classify Leibniz algebras with $$L/I\cong sl_2^1\oplus sl_2^2$$ and some conditions on ideal $$I=\mathrm{id} $$ .

In this paper, we establish necessary and sufficient conditions for boundedness of weighted p-adic Hardy operators on p-adic Morrey spaces, p-adic central Morrey spaces and p-adic $$\lambda $$ -central BMO spaces, respectively, and obtain their sharp bounds. We also give the characterization of weight functions for which the commutators generated by weighted p-adic Hardy operators and $$\lambda $$ -central BMO functions are bounded on the p-adic central Morrey spaces. This result is different from that on Euclidean spaces due to the special structure of p-adic integers.

Let \(X_n\) be a chain with n elements, and let \(\mathcal {OP}_n\) be the monoid of all orientation-preserving transformations of \(X_n\). In this paper, we investigate the nilpotent ranks of the principal factors of the semigroup \(\mathcal {OP}_n\).

A signed graph $$(G, \sigma )$$ is flow-admissible if there exists an orientation $$\tau $$ and a positive integer k such that $$(G, \sigma )$$ admits a nowhere-zero k-flow. Bouchet (J Combin Theory Ser B 34:279–292, 1983) conjectured that every flow-admissible signed graph has a nowhere-zero 6-flow. In this paper, we show that each flow-admissible signed wheel admits a nowhere-zero 4-flow if and only if G is not the specified graph. Moreover, there are infinitely many signed wheels which do not admit a nowhere-zero 3-flow. We also prove each flow-admissible signed fan admits a nowhere-zero 4-flow. Similarly, there are infinitely many signed fans which do not admit a nowhere-zero 3-flow.

A (k, d)-list assignment L of a graph is a function that assigns to each vertex v a list L(v) of at least k colors satisfying $$|L(x)\cap L(y)|\le d$$ for each edge xy. An L-coloring is a vertex coloring $$\pi $$ such that $$\pi (v) \in L(v)$$ for each vertex v and $$\pi (x) \ne \pi (y)$$ for each edge xy. A graph G is (k, d)-choosable if there exists an L-coloring of G for every (k, d)-list assignment L. This concept is known as choosability with separation. In this paper, we prove that planar graphs without 4-cycles adjacent to $$4^-$$ -cycles are (3, 1)-choosable. This is a strengthening of a result which says that planar graphs without 4-cycles are (3, 1)-choosable.

The purpose of the present paper is to prove a geometric property for analytic functions p in the open unit disk with $$p(0)=1$$ by using Nunokawa’s result, which is a generalized form of well-known Jack’s lemma. This property concerns a boundary behavior of the functions p. As the applications of the main result, we obtain a few corollaries where several sufficient conditions for p to be of the real part greater than a given number $$\beta \ (0\le \beta

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we investigate some fixed-point theorems in modular metric spaces.

A path $P$ in an edge-colored graph $G$ is called a proper path if no two adjacent edges of $P$ are colored the same, and $G$ is proper connected if every two vertices of $G$ are connected by a proper path in $G$. The proper connection number of a connected graph $G$, denoted by $pc(G)$, is the minimum number of colors that are needed to make $G$ proper connected. In this paper, we investigate the proper connection number of the complement of graph $G$ according to some constraints of $G$ itself. Also, we characterize the graphs on $n$ vertices that have proper connection number $n-2$. Using this result, we give a Nordhaus-Gaddum-type theorem for the proper connection number. We prove that if $G$ and $\overline{G}$ are both connected, then $4\le pc(G)+pc(\overline{G})\le n$, and the only graph attaining the upper bound is the tree with maximum degree $\Delta=n-2$., Comment: 12 pages

This paper deals with pairs of nonzero idempotents e and f of a third-power associative absolute valued algebra A satisfying \((ef)e= e(fe)\) and \((fe)f=f(ef)\) (pairwise flexible idempotents), and the role that they play on the structure of A. We show that if g is a nonzero idempotent of A such that the nonzero idempotents commuting with g are pairwise flexible, then the subalgebra that they generate \(B_g\) is isometrically isomorphic to \({\mathbb {R}}\), \(\mathop {{\mathbb {C}}}\limits ^{\star }\), \(\mathop {{\mathbb {H}}}\limits ^{\star }\), or \(\mathop {{\mathbb {O}}}\limits ^{\star }\). Our main theorem proves the equivalence of the following assertions: (i) for every two different nonzero idempotents e and f, the nonzero idempotents of A that commute with \((e-f)^2\) are pairwise flexible; (ii) each pair of nonzero idempotents of A generates a finite-dimensional subalgebra; and (iii) either A is isometrically isomorphic to \({\mathbb {R}}\), \({\mathbb {C}}\), \({\mathbb {H}}\), \({\mathbb {O}}\), \(\mathop {{\mathbb {C}}}\limits ^{\star }\), \(\mathop {{\mathbb {H}}}\limits ^{\star }\) or \(\mathop {{\mathbb {O}}}\limits ^{\star }\), or A contains a subalgebra B, which contains all idempotents of A and is isometrically isomorphic to the division absolute valued algebra \({\mathbb {P}}\) of the Okubo pseudo-octonions. More consequences on the structure of A related with the presence of pairwise flexible idempotents are given, among them several generalizations of some well-known theorems.

The Wiener index of a hypergraph is the sum of distances between all pairs of vertices of the hypergraph. In this paper, we first investigate the Wiener index of some special r-uniform paths and r-uniform cycles. The Wiener index of different kinds of 3-uniform paths were also considered. Then we explore the lower bound of Wiener index in a r-uniform hypergraph on n vertices with a given circumference which is the maximum length of a cycle in a hypergraph. In the last part, we propose the concept of the chemical bond-Wiener index of a graph, which is the sum of the distances between all pairs of chemical bonds of a graph, considering the removing of hydrogen atoms. The polyphenyl chains with minimum and maximum chemical bond-Wiener indices among all the polyphenyl chains with h hexagons were completely determined.

Each conjugacy class of actions of $$PGL\left( {2,{\mathbb {Z}}} \right) $$ on the projective line over a finite field $$F_q $$ denoted by $$PL\left( {F_q } \right) $$ , can be represented by a coset diagram $$D\left( {\theta ,q} \right) $$ , where $$\theta \in F_q $$ and q is a prime power. The coset diagrams are composed of fragments, and the fragments are further composed of two or more circuits at a certain common point. Professor Graham Higman raised a question: for what values of q and $$\theta $$ , can a fragment $$\gamma $$ be found in $$D\left( {\theta ,q} \right) ?$$ Mushtaq in 1983 found that the condition for the existence of a fragment in $$D\left( {\theta ,q} \right) $$ is a polynomial f in $${\mathbb {Z}}\left[ z \right] $$ . In this paper, we answer the question: how many polynomials are obtained from the fragments, composed by joining the circuits $$\left( {n_1 ,n_2 } \right) $$ and $$\left( {m_1 ,m_2 } \right) $$ , where $$n_2

Let $$\mathcal {PO}_n$$ be the semigroup of all order-preserving partial transformations on the finite set $$X_n=\{1, 2,\ldots , n\}$$ . For $$1\le r\le n-1$$ , set $$\mathcal {PO}(n, r)=\{\alpha \in \mathcal {PO}_n: |\mathop {\text{ im }}\nolimits (\alpha )|\le r\}$$ . In this paper, we investigate the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $$\mathcal {PO}(n,r)$$ . First, we completely describe the maximal regular subsemigroups of the semigroup $$\mathcal {PO}(n,r)$$ , for $$1\le r\le n-1$$ . Secondly, we show that, for $$2\le r \le n-2$$ , any maximal regular subsemigroup of the semigroup $$\mathcal {PO}(n,r)$$ is a semiband and obtain that the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $$\mathcal {PO}(n,r)$$ coincide, for $$2\le r\le n-2$$ . Finally, we obtain the complete classification of maximal regular subsemibands of the semigroup $$\mathcal {PO}_n$$ .

In this paper we investigate subordination-preserving of a certain class of integral operators, which are defined on the space of meromorphic functions. Moreover, we consider an application of the subordination Theorem to the Gauss hypergeometric function.

Hom-alternative, Hom-Malcev and Hom-Jordan superalgebras are $\mathbb{Z}_{2}$-graded generalizations of Hom-alternative, Hom-Malcev and Hom-Jordan algebras, which are Hom-type generalizations of alternative, Malcev and Jordan algebras. In this paper we prove that Hom-alternative superalgebras are Hom-Malcev-admissible and are also Hom-Jordan-admissible. Home-type generalizations of some well known identities in alternative superalgebras, including the $\mathbb{Z}_{2}$-graded Bruck-Kleinfled function are obtained., arXiv admin note: substantial text overlap with arXiv:1002.3944 by other authors

In this paper, we study the cellularity of some semigroup algebras. We show that the semigroup algebra of a U-semiabundant semigroup with Rees matrix semigroups over monoids as its principal \(\sim _U\)-factors is a cellular algebra if and only if all of the monoid algebras are cellular. We also study the cellularity of the semigroup algebra of a semilattice of Rees matrix semigroups. As consequences, we get the cellularity of super abundant semigroup algebras and complete regular semigroup algebras.

For a positive integer m, a bounded linear operator T on a Hilbert space \(\mathbb {H}\) is called an (A, m)-isometry, if \(\Theta ^{(m)}_{A}(T) =\sum _{k=0}^{m}(-1)^{m-k}{m\atopwithdelims ()k}T^{*k}AT^{k}=0\), where A is a positive (semi-definite) operator. In this paper we give a characterization of (A, m)-isometric and strict (A, m)-isometric unilateral weighted shifts in terms of their weight sequences, respectively. Moreover, we characterize (A, 2)-expansive unilateral weighted shifts (i.e. operators satisfying \(\Theta ^{(2)}_{A}(T)\le 0\)).

Let p be a prime, E be a normal subgroup of a finite group G. In this paper, we will investigate the way E embedded in G under the assumption that some p-subgroups of E are c-normal in G. We pay more attention to the p-subgroups of E with given order \(p^d\). We generalized several recent results of other scholars.

Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is $$R\setminus J(R)$$ , and two distinct vertices x and y are adjacent if and only if $$1-xy$$ is not a unit in R. In this paper, up to isomorphism, we classify the rings R whose Jacobson graphs are toroidal.

A 2-rainbow dominating function (2RDF) on a graph $$G=(V, E)$$ is a function f from the vertex set V to the set of all subsets of the set $$\{1,2\}$$ such that for any vertex $$v\in V$$ with $$f(v)=\emptyset $$ the condition $$\bigcup _{u\in N(v)}f(u)=\{1,2\}$$ is fulfilled. A 2RDF f is independent (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF f is the value $$\omega (f)=\sum _{v\in V}|f (v)|$$ . The 2-rainbow domination number $$\gamma _{r2}(G)$$ (respectively, the independent 2-rainbow domination number $$i_{r2}(G)$$ ) is the minimum weight of a 2RDF (respectively, I2RDF) on G. We say that $$\gamma _{r2}(G)$$ is strongly equal to $$i_{r2}(G)$$ and denote by $$\gamma _{r2}(G)\equiv i_{r2}(G)$$ , if every 2RDF on G of minimum weight is an I2RDF. In this paper, we provide a constructive characterization of trees T with $$\gamma _{r2}(T)\equiv i_{r2}(T)$$ .

The Pasch axiom is a strong geometric property which was noted and discussed even from the period of Euclid. Modern geometers followed the study of the Pasch axiom, and in the context of axiomatic convexity, this axiom was widely studied by pioneers in convexity theory. Connected graphs for which the geodesic interval function I satisfies the Pasch axiom were characterized by Chepoi in 1994. In this paper, we characterize all graphs for which the induced path transit function satisfies the Pasch axiom. Among arbitrary transit functions R that satisfy the Pasch axiom, we identify the induced path transit function of the underlying graph $$G_R$$ by a set of axioms.

A graph is equitably k-colorable if G has a proper vertex k-coloring such that the sizes of any two color classes differ by at most one. Chen, Lih and Wu conjectured that any connected graph G with maximum degree $$\Delta $$ distinct from the odd cycle, the complete graph $$K_{\Delta +1}$$ and the complete bipartite graph $$K_{\Delta ,\Delta }$$ are equitably m-colorable for every $$m\ge \Delta $$ . Let $${\mathcal {G}}_k$$ be the class of graphs G such that $$e(G')\le k (v(G')-2)$$ for every subgraph $$G'$$ of G with order at least 3. In this paper, it is proved that any graph in $${\mathcal {G}}_4$$ with maximum degree $$\Delta \ge 17$$ is equitably m-colorable for every $$m\ge \Delta $$ . As corollaries, we confirm Chen–Lih–Wu Conjecture for 1-planar graphs, 3-degenerate graphs and graphs with maximum average degree less than 6, provided that $$\Delta \ge 17$$ .

The harmonic index of a graph G is defined as the sum of weights $$\frac{2}{d(u)+d(v)}$$ over all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. Let $$\mathscr {T}(n,\gamma )$$ and $$\mathscr {C}_n^\pi $$ be the sets of trees of order n with domination number $$\gamma $$ and connected graphs of order n with degree sequence $$\pi $$ , respectively. In this paper, the tree with minimum harmonic index among trees in $$\mathscr {T}(n,\gamma )$$ is determined, as well as the trees with maximum harmonic index among trees in $$\mathscr {T}(n,\gamma )$$ when $$\gamma =2,\lceil \frac{n}{3}\rceil $$ . Moreover, we show that among graphs in $$\mathscr {C}_n^\pi $$ , the maximum harmonic index is attained by a BFS graph.

An edge-coloring of a connected graph is a monochromatically-connecting coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices, which was introduced by Caro and Yuster. Let mc(G) denote the maximum number of colors used in an MC-coloring of a graph G. Note that an MC-coloring does not exist if G is not connected, in which case we simply let \(mc(G)=0\). In this paper, we characterize all connected graphs of size m with \(mc(G)=1, 2, 3, 4\), \(m-1\), \(m-2\) and \(m-3\), respectively. We use G(n, p) to denote the Erdős-Renyi random graph model, in which each of the \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) pairs of vertices appears as an edge with probability p independent from other pairs. For any function f(n) satisfying \(1\le f(n)

In this paper, we firstly consider how the normalized Laplacian spectral radius of a non-bipartite graph behaves by several graph operations. As applications of the result, the largest normalized Laplacian spectral radius of non-bipartite unicyclic graphs with fixed order and girth is determined. Moreover, the largest normalized Laplacian spectral radius among non-bipartite unicyclic graphs with fixed girth and order is also determined. The maximizer is the tadpole graph with the same (odd) girth.