Search

In this paper, we prove that if a nonconstant finite order meromorphic function f and its n-th order difference operator $$\Delta ^n_{\eta }f$$ share $$a_1,$$ $$a_2,$$ $$a_3$$ CM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, and $$a_1,$$ $$a_2,$$ $$a_3$$ are three distinct finite complex values, then $$f(z)=\Delta ^n_{\eta }f(z)$$ for each $$z\in \mathbb {C}.$$ The main results in this paper improve and extend many known results concerning a conjecture posed by Chen and Yi in 2013.

In this paper, we study a dynamic coloring of the vertices of a graph G that starts with an initial subset S of colored vertices, with all remaining vertices being uncolored. At each discrete time interval, a colored vertex with exactly one uncolored neighbor forces this uncolored neighbor to be colored. The initial set S is a zero forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set of G. In this paper, we prove that if G is a connected, cubic, claw-free graph of order $$n \ge 6$$, then $$Z(G) \le \alpha (G) + 1$$ where $$\alpha (G)$$ denotes the independence number of G. Further we prove that if $$n \ge 10$$, then $$Z(G) \le \frac{1}{3}n + 1$$. Both bounds are shown to be best possible.

In a recent paper, Das introduced the graph $$\mathcal {I}n(\mathbb {V})$$ , called subspace inclusion graph on a finite dimensional vector space $$\mathbb {V}$$ , where the vertex set is the collection of nontrivial proper subspaces of $$\mathbb {V}$$ and two vertices are adjacent if one is properly contained in another. Das studied the diameter, girth, clique number and chromatic number of $$\mathcal {I}n(\mathbb {V})$$ when the base field is arbitrary, and he also studied some other properties of $$\mathcal {I}n(\mathbb {V})$$ when the base field is finite. At the end of the above paper, the author posed the open problem of determining the automorphisms of $$\mathcal {I}n(\mathbb {V})$$ . In this paper, we give the answer to the open problem.

In this paper, we classify all isolated, completely isolated, and (left/right) convex subsemigroups of $${\mathcal {C}}_{n}$$ , the semigroup of all order-preserving and decreasing transformations on $$X_{n}=\{1,\ldots ,n\}$$ under its natural order. Moreover, we find the rank of each convex subsemigroup of $${\mathcal {C}}_{n}$$ .

In this paper, we study ad-nilpotent elements of semiprime rings R with involution $$*$$ whose indices of ad-nilpotence differ on $${{\,\mathrm{Skew}\,}}(R,*)$$ and R. The existence of such an ad-nilpotent element a implies the existence of a GPI of R, and determines a big part of its structure. When moving to the symmetric Martindale ring of quotients $$Q_m^s(R)$$ of R, a remains ad-nilpotent of the original indices in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ and $$Q_m^s(R)$$ . There exists an idempotent $$e\in Q_m^s(R)$$ that orthogonally decomposes $$a=ea+(1-e)a$$ and either ea and $$(1-e)a$$ are ad-nilpotent of the same index (in this case the index of ad-nilpotence of a in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ is congruent with 0 modulo 4), or ea and $$(1-e)a$$ have different indices of ad-nilpotence (in this case the index of ad-nilpotence of a in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ is congruent with 3 modulo 4). Furthermore, we show that $$Q_m^s(R)$$ has a finite $${\mathbb {Z}}$$ -grading induced by a $$*$$ -complete family of orthogonal idempotents and that $$eQ_m^s(R)e$$ , which contains ea, is isomorphic to a ring of matrices over its extended centroid. All this information is used to produce examples of these types of ad-nilpotent elements for any possible index of ad-nilpotence n.

If a graph G is chordal, Ha and Van Tuyl proved that $${{\,\mathrm{reg}\,}}(G) = \nu _0(G)$$ , where $$\nu _0(G)$$ is the induced matching number of G. In this paper, we produce a lower bound for $${{\,\mathrm{reg}\,}}(G)$$ which coincides with the result of Ha and Van Tuyl in the case of chordal graphs. We also consider the case G is vertex-decomposable.

Given a graph G and an edge $$e=xy$$ in G, let $$n_x(e)$$ and $$n_y(e)$$ be the number of vertices that have the distance to x less than that to y, and the number of vertices that have the distance to y less than that to x, respectively. The contribution of e is defined as $$|n_x(e)-n_y(e)|$$ . The Mostar index of G is the sum of all edge contributions in G. A segment in a given tree T is a path, each of whose inner vertices has degree exactly 2, and none of whose two ends has the degree 2. The segment sequence of T is the length sequence of all segments in T. In this paper, we focus on the tree set with a fixed segment sequence, and the tree set with a fixed size together with a fixed segment number. We completely determine the graphs with the greatest Mostar index among the two sets, respectively. The graphs with the least Mostar index among the second set are also identified.

In this paper, we study the following nonlinear fractional Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b\int _{\mathbb {R}^{3}}|(-\Delta )^{\frac{s}{2}}u|^{2}\mathrm{d}x\right) (-\Delta )^{s}u+\lambda V(x)u=|u|^{p-2}u, &{} {\mathrm{in }}\mathbb {R}^3, \\ u\in H^{s}(\mathbb {R}^3), \end{array} \right. \end{aligned}$$ where $$s\in (\frac{3}{4}, 1)$$ , $$a, b, \lambda $$ are positive parameters, $$2

In this paper, we first prove that the connected cubic edge-transitive bi-Cayley graphs over a generalized dihedral group have girth 6. Using this, a complete classification is given of these graphs.

In this paper, we consider the following critical p-biharmonic equation involving Hardy potential $$\begin{aligned} \begin{aligned} \Delta _p^2u-\Delta _qu-\mu \frac{|u|^{p-2}u}{|x|^{2p}}=|u|^{p_{*}-2}u,\quad {{x\in \mathbb {R}^{N}}}, \end{aligned} \end{aligned}$$ where $$2\leqslant p

Given a graph G, the burning number of G is the smallest integer k for which there are vertices $$x_1, x_2,\ldots ,x_k$$ such that $$(x_1,x_2,\ldots ,x_k)$$ is a burning sequence of G. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A t-unicyclic graph is a unicycle graph in which the unique vertex in the graph with degree greater than two has degree $$ t + 2 $$ . In this paper, we first present the bounds for the burning number of t-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burning number of all t-unicyclic graphs for $$t\le 2$$ .

In this paper, several new estimates of the minimum H-eigenvalue for weakly irreducible nonsingular $$\mathcal {M}$$ -tensors, including the new Brauer-type estimates and the new S-type estimates, are derived. It is proved that the new estimates are tighter than some existing ones and numerical examples are given to verify this fact. The other main result of this paper is to provide a sharper Ky Fan-type theorem which is better than the original Ky Fan theorem for the nonsingular $$\mathcal {M}$$ -tensors.

As the generalization of the digraph, the directed hypergraph (dihypergraph), denoted by $${H}$$ , is considered in this paper. First, we introduce the notions of the zeta function of $${H}$$ and dihypergraph coverings over $${H}$$ . Next, several expressions for the zeta function of $${H}$$ are given, and all dihypergraph coverings over $${H}$$ are generated by permutation voltage assignments on the incidence bipartite digraph or the edge-colored digraph of $${H}$$ . Final, we derive two explicit decomposition formulae for the zeta function of any dihypergraph covering $$\overline{{H}}$$ over $${H}$$ , which turn out that the zeta function of $${H}$$ divides the zeta function of $$\overline{{H}}$$ .

This paper deals with the existence of positive radial solutions to the iterative system of nonlinear elliptic equations of the form $$\begin{aligned} \begin{aligned} \triangle {\mathtt {z}_{\mathtt {j}}}+\frac{(\mathtt {N}-2)^2r_0^{2\mathtt {N}-2}}{\vert x\vert ^{2\mathtt {N}-2}}\mathtt {z}_\mathtt {j}+\varphi (\vert x\vert )\mathtt {g}_{\mathtt {j}}(\mathtt {z}_{\mathtt {j}+1})=0,~\mathtt {R}_12,$$ $$0

Structured frames in $$L^2(\mathbb R)$$ , such as wavelet and Gabor frames, have been extensively studied. But the analysis on $$L^2(\mathbb R_+)$$ with $$\mathbb R_+=(0,\infty )$$ has been rarely reported. It is because $$\mathbb R$$ is a group under addition but $$\mathbb R_+$$ is not. This results in $$L^2(\mathbb R_+)$$ admitting no traditional wavelet or Gabor frames. Recently, the concept of $${F_{a}{\text{-frame }}}$$ in $$L^2(\mathbb R_+)$$ was introduced and studied. In this paper, we present some identities and inequalities for $${F_{a}{\text{-frame }}}$$ sequences in $$L^2(\mathbb R_+)$$ .

A permutation group G acting on a set $$\Omega $$ induces a permutation group on the power set $${\mathscr {P}}(\Omega )$$ (the set of all subsets of $$\Omega $$ ). Let G be a finite permutation group of degree n and s(G) denote the number of orbits of G on $${\mathscr {P}}(\Omega )$$ . It is an interesting problem to determine the lower bound $$\inf \left( \frac{\log _2 s(G)}{n}\right) $$ over all groups G that do not contain any alternating group $$\mathrm{A}_\ell $$ (where $$\ell >t$$ for some fixed $$t\geqslant 4)$$ as a composition factor. The second author obtained the answer for the case $$t=4$$ in Yang (J Algebra Appl 19:2150005, 2020). In this paper, we continue this investigation and study the cases when $$t\geqslant 5$$ , and give the explicit lower bounds $$\inf \left( \frac{\log _2 s(G)}{n}\right) $$ for each positive integer $$5\leqslant t\leqslant 166$$ .

In this paper, we give a new product formula : $$\begin{aligned} {\mathsf {H}}^t(E) {\mathsf {H}}^s(F)\le c_1 {\mathsf {H}}^{t+s}(E\times F)\le & {} c_2 {\mathsf {H}}^s(E) {\mathsf {P}}^t(F)\\\le & {} c_3 {\mathsf {P}}^{t+s}(E\times F) \le c_4 {\mathsf {P}}^s(E) {\mathsf {P}}^t(F). \end{aligned}$$ where $$E\subseteq \mathbb {R}^d$$ , $$F\subseteq \mathbb {R}^l$$ , $$t,s\ge 0$$ and $$ {\mathsf {H}}^t$$ and $${\mathsf {P}}^s$$ denote, respectively, the lower and upper Hewitt–Stromberg measures. Using these inequalities, we give lower and upper bounds for the lower and upper Hewitt–Stromberg dimensions $${\mathsf {b}}(E\times F)$$ and $${\mathsf {B}}(E\times F)$$ in terms of the Hewitt–Strombeg dimensions of E and F.

In this paper, we investigate the following Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+u+\phi u=f(u), &{}\mathrm {in}\ \mathbb {R}^{3},\\ \displaystyle -\Delta \phi =u^2,&{}\mathrm {in}\ \mathbb {R}^{3}, \end{array}\right. \end{aligned}$$ where $$f\in C(\mathbb {R},\mathbb {R})$$ , and f satisfies that $$\frac{1}{3}f(t)t\ge F(t)>0$$ with $$t\in \mathbb {R}\backslash \{0\}$$ . We construct a Palais–Smale sequence that is relevant to Pohožaev identity to obtain infinitely many high energy radial solutions for this system.

Let G be a simple, connected graph and let A(G) be the adjacency matrix of G. If D(G) is the diagonal matrix of the vertex degrees of G, then for every real $$\alpha \in [0,1]$$ , the matrix $$A_{\alpha }(G)$$ is defined as $$A_{\alpha }(G) = \alpha D(G) + (1- \alpha ) A(G).$$ The eigenvalues of the matrix $$A_{\alpha }(G)$$ form the $$A_{\alpha }$$ -spectrum of G. Let $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ denote the subdivision-vertex join, subdivision-edge join, R-vertex join and R-edge join of two graphs $$G_1$$ and $$G_2$$ , respectively. In this paper, we compute the $$A_{\alpha }$$ -spectra of $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ for a regular graph $$G_1$$ and an arbitrary graph $$G_2$$ in terms of their $$A_{\alpha }$$ -eigenvalues. As an application of these results, we construct infinitely many pairs of $$A_{\alpha }$$ -cospectral graphs.

A graph is NIC-planar if it can be drawn in the plane so that there is at most one crossing per edge and two pairs of crossing edges share at most one common end vertex. A (p, 1)-total k-labelling of a graph G is a function f from $$V(G)\cup E(G)$$ to the color set $$\{0,1,\ldots ,k\}$$ such that $$|f(x)-f(y)|\ge 1$$ if $$xy\in E(G)$$ , $$|f(e_1)-f(e_2)|\ge 1$$ if $$e_1$$ and $$e_2$$ are two adjacent edges in G, and $$|f(x)-f(e)|\ge p$$ if a vertex x is incident with the edge e. The minimum k such that G has a (p, 1)-total k-labelling is the (p, 1)-total labelling number of G. In this paper, we prove that the (p, 1)-total labelling number ( $$p\ge 2$$ ) of every NIC-planar graph G is at most $$\Delta (G)+2p-2$$ provided that $$\Delta (G)\ge 6p+4.$$

Let $$z_0$$ and $$w_0$$ be given points in the open unit disk $$\mathbb {D}$$ with $$|w_0| < |z_0|$$ , and $$\mathcal {H}_0$$ be the class of all analytic self-maps f of $$\mathbb {D}$$ normalized by $$f(0)=0$$ . In this paper, we establish the third-order Dieudonne’s Lemma and apply it to explicitly determine the variability region $$\{f'''(z_0): f\in \mathcal {H}_0,f(z_0) =w_0, f'(z_0)=w_1\}$$ for given $$z_0,w_0,w_1$$ and give the form of all the extremal functions.

In this paper, we consider a nonlinear Schrodinger equation involving the fractional Laplacian with Dirichlet condition: $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{\frac{\alpha }{2}}u+A(x)u=f(x,u,\nabla u) \ \text{ in } \ \Omega ,\\ u>0, \ \text{ in }\ \Omega ; \ u\equiv 0, \ \text{ in }\ \mathbb R^n\backslash \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a domain (bounded or unbounded) in $$\mathbb R^n$$ which is convex in $$x_1$$ -direction. By using some ideas of maximum principle and the direct moving plane method, we prove that the solutions are strictly increasing in $$x_1$$ -direction in the left half domain of $$\Omega $$ . Symmetry of some solutions are also proved. Meanwhile, we obtain a Liouville type theorem on the half space $$\mathbb R^n_+$$ .

Let f, g be positive integers such that $$f>g$$ , $$\gcd (f,g)=1$$ and $$f\not \equiv g \pmod 2$$ . In 1956, L. Jeśmanowicz conjectured that, for any positive integer n, the equation $$(*)$$ $$\left( (f^2-g^2)n\right) ^x+(2fgn)^y=\left( (f^2+g^2)n\right) ^z$$ has only the positive integer solution $$(x,y,z)=(2,2,2)$$ . This is a far from solved problem in number theory. In 1965, V.A. Dem’janenko claimed that if $$n=1$$ and $$f=g+1$$ , then $$(*)$$ has only the positive integer solution $$(x,y,z)=(2,2,2)$$ . This result solves an important case of Jeśmanowicz’ conjecture. In this paper, on the one hand, using some properties on the representation of integers by binary quadratic primitive forms, we give a new and elementary proof for Dem’janenko’s result; on the other hand, using the Baker method and its p-adic form, we prove that if $$n>1$$ , $$f=g+1$$ and $$g=2^r$$ , where r is a positive integer with $$r\ge 80$$ and $$r+1$$ is an odd prime, then $$(*)$$ has only the positive integer solution $$(x,y,z)=(2,2,2)$$ . It can thus be seen that if $$n>1$$ , then there are infinitely many pairs (f, g) with $$f=g+1$$ which make Jeśmanowicz’ conjecture is true.

Different types of domination on the Sierpinski graphs $$S(K_n,t)$$ will be studied in this paper. More precisely, we propose a minimal dominating set for each $$S(K_n,t)$$ so that the exact values of its Roman domination numbers and double Roman domination numbers are given. As applications, some previous bounds and results are confirmed to be tight and further generalized.

Let G be an edge-colored connected graph. A path P in G is called $$\ell $$ -rainbow if no two edges of the same color have fewer than $$\ell $$ edges between them on P. The graph G is called $$(k,\ell )$$ -rainbow connected if every pair of distinct vertices of G are connected by k pairwise internally vertex-disjoint $$\ell $$ -rainbow paths in G. For a k-connected graph G, the minimum number of colors needed to make G $$(k,\ell )$$ -rainbow connected is called the $$(k,\ell )$$ -rainbow connection number of G and denoted by $$rc_{k,\ell }(G)$$ . In this paper, we prove that $$rc_{1,2}(G)\le 5$$ for any 2-connected graph G and provide a counter example which overturns a conjecture posed by Chartrand et al. in 2016. Considering graph operations, we study the (1, 2)-rainbow connection numbers for the join, the Cartesian, strong and lexicographic products of graphs, giving the sharp upper bounds for nearly all graph products. In addition, we find some basic properties of the $$(k,\ell )$$ -rainbow connection number and determine the values of $$rc_{1,\ell }(G)$$ where G is a cube or a permutation graph of a nontrivial traceable graph.

Let $$w=(w_0,w_1, \dots ,w_l)$$ be a vector of nonnegative integers such that $$ w_0\ge 1$$ . Let G be a graph and N(v) the open neighbourhood of $$v\in V(G)$$ . We say that a function $$f: V(G)\longrightarrow \{0,1,\dots ,l\}$$ is a w-dominating function if $$f(N(v))=\sum _{u\in N(v)}f(u)\ge w_i$$ for every vertex v with $$f(v)=i$$ . The weight of f is defined to be $$\omega (f)=\sum _{v\in V(G)} f(v)$$ . Given a w-dominating function f and any pair of adjacent vertices $$v, u\in V(G)$$ with $$f(v)=0$$ and $$f(u)>0$$ , the function $$f_{u\rightarrow v}$$ is defined by $$f_{u\rightarrow v}(v)=1$$ , $$f_{u\rightarrow v}(u)=f(u)-1$$ and $$f_{u\rightarrow v}(x)=f(x)$$ for every $$x\in V(G){\setminus }\{u,v\}$$ . We say that a w-dominating function f is a secure w-dominating function if for every v with $$f(v)=0$$ , there exists $$u\in N(v)$$ such that $$f(u)>0$$ and $$f_{u\rightarrow v}$$ is a w-dominating function as well. The (secure) w-domination number of G, denoted by ( $$\gamma _{w}^s(G)$$ ) $$\gamma _{w}(G)$$ , is defined as the minimum weight among all (secure) w-dominating functions. In this paper, we show how the secure (total) domination number and the (total) weak Roman domination number of lexicographic product graphs $$G\circ H$$ are related to $$\gamma _w^s(G)$$ or $$\gamma _w(G)$$ . For the case of the secure domination number and the weak Roman domination number, the decision on whether w takes specific components will depend on the value of $$\gamma _{(1,0)}^s(H)$$ , while in the case of the total version of these parameters, the decision will depend on the value of $$\gamma _{(1,1)}^s(H)$$ .

The Wiener index is one of the most widely studied parameters in chemical graph theory. It is defined as the sum of the lengths of the shortest paths between all unordered pairs of vertices in a given graph. In 1991, Soltes posed the following problem regarding the Wiener index: Find all graphs such that its Wiener index is preserved upon removal of any vertex. The problem is far from being solved, and to this day, only one graph with such property is known: the cycle graph on 11 vertices. In this paper, we solve a relaxed version of the problem, proposed by Knor et al. in 2018. For a given k, the problem is to find (infinitely many) graphs having exactly k vertices such that the Wiener index remains the same after removing any of them. We call these vertices good vertices, and we show that there are infinitely many cactus graphs with exactly k cycles of length at least 7 that contain exactly 2k good vertices and infinitely many cactus graphs with exactly k cycles of length $$c \in \{5,6\}$$ that contain exactly k good vertices. On the other hand, we prove that G has no good vertex if the length of the longest cycle in G is at most 4.

The connective eccentricity index is a degree-distance-based graph invariant, while the modified second Zagreb index is a degree-based graph invariant. The Parikh word representable graph is a new class of graphs G(w), which corresponds to words w that are finite sequence of symbols. In this paper, we first present explicit formulas for the connective eccentricity index and modified second Zagreb index of the Parikh word representable graphs corresponding to binary core words of the form aub over a binary alphabet $$\{a,\,b\}$$ , respectively. Then, we investigate the relationship between the connective eccentricity index and modified second Zagreb index and obtain some comparative results about these two indices.

Given a finite graph G, the maximum length of a sequence $$(v_1,\ldots ,v_k)$$ of vertices in G such that each $$v_i$$ dominates a vertex that is not dominated by any vertex in $$\{v_1,\ldots ,v_{i-1}\}$$ is called the Grundy domination number, $$\gamma _\mathrm{gr}(G)$$ , of G. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that $$\gamma _\mathrm{gr}(G) \ge \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}$$ holds for every connected k-regular graph of order n different from $$K_{k+1}$$ and $$\overline{2C_4}$$ . The bound in the case $$k=3$$ reduces to $$\gamma _\mathrm{gr}(G)\ge \frac{n}{2}$$ , and we characterize the connected cubic graphs with $$\gamma _\mathrm{gr}(G)=\frac{n}{2}$$ . If G is different from $$K_4$$ and $$K_{3,3}$$ , then $$\frac{n}{2}$$ is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound.

Hamming graph is known to be an important class of graphs, and it is a challenge to obtain algorithms that recognize whether a given graph G is a Hamming graph. Let G be a group and $$S\subseteq G$$ be a nonempty subset of G. The Cayley graph with respect to S is a graph whose vertex set is G and arcset is the set of pairs (u, v) such that $$v=su$$ for some element $$s\in S$$ . This graph is denoted by $$\mathrm{Cay}(G,S)$$ .Let $$R=\oplus _{i}R_{i}$$ be a graded ring, S be the set of homogeneous elements of R, $$S'$$ a subset of S, and $$S''=\oplus _{i\ge k}R_{i}$$ . In this paper, with a different view, we study $$\mathrm{Cay}(R, S')$$ as a generalization of $$\mathrm{Cay}(R, S)$$ to obtain a new point of view to study Cartesian products of complete graphs (Hamming graph). In particular, we show that any Hamming graph over sets of prime power sizes is isomorphic to $$\mathrm{Cay}(R,S')$$ for some graded ring R and a subset $$S'\subseteq S$$ . Also we study $$\mathrm{Cay}(R, S'')$$ as another Cayley graph over graded rings and obtain relations between this graph and total, cototal and counit graphs.

The commuting graph $$\varDelta (G)$$ of a finite non-abelian group G is a simple graph with vertex set G, and two distinct vertices x, y are adjacent if $$xy = yx$$ . In this paper, first we discuss some properties of $$\varDelta (G)$$ . We determine the edge connectivity and the minimum degree of $$\varDelta (G)$$ and prove that both are equal. Then, other graph invariants, namely: matching number, clique number, boundary vertex, of $$\varDelta (G)$$ are studied. Also, we give necessary and sufficient condition on the group G such that the interior and center of $$\varDelta (G)$$ are equal. Further, we investigate the commuting graph of the semidihedral group $$SD_{8n}$$ . In this connection, we discuss various graph invariants of $$\varDelta (SD_{8n})$$ including vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of $$\varDelta (SD_{8n})$$ .

The independence polynomial of a graph G is $$I(G;x)=\sum _{k=0}^{\alpha (G)} s_{k}\cdot x^{k}$$ , where $$s_{k}$$ and $$\alpha (G)$$ denote the number of independent sets of cardinality k and the independence number of G, respectively. We say that a cycle is a $$\tilde{3}$$ -cycle if its length is divisible by 3, otherwise a non- $$\tilde{3}$$ -cycle. Define $$\phi (G)$$ to be the decycling number of G. Engstrom proved that $$|I(G;-1)| \le 2^{\phi (G)}$$ for any graph G. In this paper, we first prove that $$|I(G;-1)|\le 2^{\beta (G)}-\beta (G)$$ for graphs with non- $$\tilde{3}$$ -cycles, where $$\beta (G)$$ is the cyclomatic number of G. Infinitely many examples show that there do exist graphs satisfying $$\beta (G)=\phi (G)$$ and containing non- $$\tilde{3}$$ -cycles. In such a case, it improves the Engstrom’s result. Furthermore, $$|I(G;-1)|\le 2^{k}$$ provided that all cycles of G are pairwise disjoint, where k is the number of $$\tilde{3}$$ -cycles of G. This provides a new perspective on estimating the independence polynomial at -1 of many special graphs. In the case G contains no vertices of degree 1, $$|I(G;-1)|\le 2^{\beta (G)}-1$$ if $$\beta (G)\ge 2$$ , and $$|I(G;-1)|\le 2^{\beta (G)-1}$$ if G contains a non- $$\tilde{3}$$ -cycle.

The class of $$\{P_1,P_2\}$$ -Nekrasov matrices, defined in terms of permutation matrices $$P_1$$ and $$P_2$$ , is a generalization of the well-known class of Nekrasov matrices. In this paper, some computable error bounds for linear complementarity problems (LCPs) of $$\{P_1,P_2\}$$ -Nekrasov matrices are given, which depend only on the entries of the involved matrices and can be used to obtain the perturbation bounds of $$\{P_1,P_2\}$$ -Nekrasov matrices LCPs. Besides, some sufficient conditions ensuring that the subdirect sum of $$\{P_1,P_2\}$$ -Nekrasov matrices lies in the same class are also provided.

The k-rainbow index $$rx_k(G)$$ of a connected graph G was introduced in 2010. As a natural counterpart of the k-rainbow index, the concept of k-vertex-rainbow index $$rvx_k(G)$$ of a connected graph G was introduced in 2016. In this paper, we mainly investigate the k-vertex-rainbow index of a connected graph G and obtain some sharp bounds for $$rvx_k(G)$$ . In particular, by using Lovasz Local Lemma and dominating sets, we prove that a connected graph G with n vertices and minimum degree $$\delta $$ has $$rvx_3(G) < \frac{11n}{\delta }+16$$ . In general, the investigation of k-vertex-rainbow index of a graph may be harder than that of k-rainbow index of a graph. For example, it is noticed that $$rx_2(G)\le rx_3(G)\le \cdots \le rx_n(G)$$ for every connected graph G of order n, but we show that this monotonicity of vertex-rainbow index does not hold in some graph classes by illustrating a counterexample which is a cycle of order n. As a byproduct, we get the exact values of k-vertex-rainbow index for a cycle of order n when $$k=3,n-4,n-3,n-2,n-1$$ and n, respectively.

Let $$K_i$$ be plane convex bodies with the perimeters $$L_{K_i}$$ and areas $$A_{K_i}$$ for $$i=1,2$$ , respectively. In this paper, the lower bounds of the mixed isoperimetric deficit $$\Delta _{K_1,K_2}=L_{K_1}^2L_{K_2}^2-16\pi ^2 A_{K_1}A_{K_2}$$ are obtained; these bounds involve the areas enclosed by the evolute and the Wigner caustic, Hausdorff and $$L_2$$ distances between two convex bodies.

Let X, Y be independent continuous univariate random variables with unknown distributions. Suppose we observe two independent random samples $$X'_1, \ldots , X'_n$$ and $$Y'_1, \ldots , Y'_m$$ from the distributions of $$X' = X+\zeta $$ and $$Y'=Y+\eta $$ , respectively. Here $$\zeta $$ , $$\eta $$ are random noises and have known distributions. This paper is devoted to an estimation for unknown cumulative distribution function (cdf) $$F_{X+Y}$$ of the sum $$X+Y$$ on the basis of the samples. We suggest a nonparametric estimator of $$F_{X+Y}$$ and demonstrate its consistency with respect to the root mean squared error. Some upper and minimax lower bounds on convergence rate are derived when the cdf’s of X, Y belong to Sobolev classes and when the noises are Fourier-oscillating, supersmooth and ordinary smooth, respectively. Particularly, if the cdf’s of X, Y have the same smoothness degrees and $$n=m$$ , our estimator is minimax optimal in order when the noises are Fourier-oscillating as well as supersmooth. A numerical example is also given to illustrate our method.

Let R be a noncommutative prime ring of characteristic different from 2 and 3, C the extended centroid of R, F and G two generalized derivations of R, d a nonzero derivation of R and $$f(x_1,\ldots ,x_n)$$ a multilinear polynomial over C. Suppose that I is a nonzero ideal of R and $$f(I)=\{f(x_1,\ldots ,x_n)| x_1,\ldots ,x_n\in I\}$$ . If $$f(x_1,\ldots ,x_n)$$ is not central valued on R and $$\begin{aligned} d(F^2(u)u-G(u^2))=0 \end{aligned}$$ for all $$u\in f(I)$$ , then all possible forms of the maps F, G and d are described in the present paper.

In this work, we estimate the bounds for the logarithmic coefficients $$\gamma _n$$ of some well-known classes like $${\mathcal {F}}(c)$$ for $$c\in (0,0.656]\cup \{2\}$$ . The best bound obtained for the logarithmic coefficient $$|\gamma _5|$$ is sharp for $$c=2$$ . In a special case, it is important to note that we obtain the bounds for the logarithmic coefficients $$\gamma _n$$ of the convex functions of order $$\alpha $$ for some $$\alpha $$ . It is worthwhile mentioning that the given bounds would generalize some of the previous papers.

In 2011, Gross derived an $$O(n^2)$$ -time algorithm to calculate the genus distribution of a given cubic Halin graph. In this paper, with the help of the overlap matrix, we obtain a recurrence relation for the Euler-genus polynomials of cubic caterpillar-Halin graphs. Explicit formulas for the embeddings of the cubic caterpillar-Halin graphs into surfaces with Euler-genus 0, 1 and 2 are also obtained.

In this paper, we study ad-nilpotent elements in Lie algebras arising from semiprime associative rings R free of 2-torsion. With the idea of keeping under control the torsion of R, we introduce a more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a only technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows us to be more precise when setting the torsion inside the ring R in order to describe its ad-nilpotent elements. If R is a semiprime ring and $$a\in R$$ is a pure ad-nilpotent element of R of index n with R free of t and $$\left( {\begin{array}{c}n\\ t\end{array}}\right) $$ -torsion for $$t=[\frac{n+1}{2}]$$ , then n is odd and there exists $$\lambda \in C(R)$$ such that $$a-\lambda $$ is nilpotent of index t. If R is a semiprime ring with involution $$*$$ and a is a pure ad-nilpotent element of $${{\,\mathrm{Skew}\,}}(R,*)$$ free of t and $$\left( {\begin{array}{c}n\\ t\end{array}}\right) $$ -torsion for $$t=[\frac{n+1}{2}]$$ , then either a is an ad-nilpotent element of R of the same index n (this may occur if $$n\equiv 1,3 \,(\text {mod } 4)$$ ) or R is a nilpotent element of R of index $$t+1$$ , and R satisfies a nontrivial GPI (this may occur if $$n\equiv 0,3 \,(\text {mod } 4)$$ ). The case $$n\equiv 2 \,(\text {mod } 4)$$ is not possible.

For a connected graph G(V, E), a vertex $$w\in V(G)$$ strongly resolves two vertices $$u, v \in V(G)$$ if there exists a shortest $$u-w$$ path containing v or a shortest $$v-w$$ path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is a simple graph with the vertex set $$Z(R)^*=Z(R){\setminus }\{0\}$$ , and two distinct vertices x and y are adjacent if and only if $$ann_R(xy)\ne ann_R(x)\cup ann_R(y)$$ . In this paper, we study the strong metric dimension of annihilator graphs associated with commutative rings and some strong metric dimension formulae for annihilator graphs are given.

For an edge-colored graph $G$, a set $F$ of edges of $G$ is called a \emph{proper cut} if $F$ is an edge-cut of $G$ and any pair of adjacent edges in $F$ are assigned by different colors. An edge-colored graph is \emph{proper disconnected} if for each pair of distinct vertices of $G$ there exists a proper edge-cut separating them. For a connected graph $G$, the \emph{proper disconnection number} of $G$, denoted by $pd(G)$, is the minimum number of colors that are needed in order to make $G$ proper disconnected. In this paper, we first give the exact values of the proper disconnection numbers for some special families of graphs. Next, we obtain a sharp upper bound of $pd(G)$ for a connected graph $G$ of order $n$, i.e, $pd(G)\leq \min\{ \chi'(G)-1, \left \lceil \frac{n}{2} \right \rceil\}$. Finally, we show that for given integers $k$ and $n$, the minimum size of a connected graph $G$ of order $n$ with $pd(G)=k$ is $n-1$ for $k=1$ and $n+2k-4$ for $2\leq k\leq \lceil\frac{n}{2}\rceil$., Comment: 14 pages

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. The authors introduced and studied $\xi$-$\mathcal{G}$projective and $\xi$-$\mathcal{G}$injective in \cite{HZZ}. In this paper, we discuss Gorenstein homological dimensions for extriangulated categories. More precisely, we first give some characterizations of $\xi$-$\mathcal{G}$projective dimension by using derived functors on $\mathcal{C}$. Second, let $\mathcal{P}(\xi)$ (resp. $\mathcal{I}(\xi)$) be a generating (resp. cogenerating) subcategory of $\mathcal{C}$. We show that the following equality holds under some assumptions: $$\sup\{\xi\textrm{-}\mathcal{G}{\rm pd}M \ | \ \textrm{for} \ \textrm{any} \ M\in{\mathcal{C}}\}=\sup\{\xi\textrm{-}\mathcal{G}{\rm id}M \ | \ \textrm{for} \ \textrm{any} \ M\in{\mathcal{C}}\},$$ where $\xi\textrm{-}\mathcal{G}{\rm pd}M$ (resp. $\xi\textrm{-}\mathcal{G}{\rm id}M$) denotes $\xi$-$\mathcal{G}$projective (resp. $\xi$-$\mathcal{G}$injective) dimension of $M$. As an application, our main results generalize their work by Bennis-Mahdou and Ren-Liu. Moreover, our proof is not far from the usual module or triangulated case., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1906.10989

In this paper, we are interested in the brush number of a graph—a concept introduced by McKeil, Messinger, Nowakowski and Pralat. Our main aim in this paper is to study the brush number of the Cartesian product of a tree with a path, and a tree with a cycle. Based on an optimal cleaning process of a tree, we describe cleaning processes that give upper bounds to the brush number of $$T\times P_n$$ and $$T\times C_n$$ . We also show that these bounds are tight if T is a star, a double star and some generalisation of a star.

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.

A set of vertices W of a graph G is a resolving set if every vertex of G is uniquely determined by its vector of distances to W. In this paper, the Maker–Breaker resolving game is introduced. The game is played on a graph G by Resolver and Spoiler who alternately select a vertex of G not yet chosen. Resolver wins if at some point the vertices chosen by him form a resolving set of G, whereas Spoiler wins if the Resolver cannot form a resolving set of G. The outcome of the game is denoted by o(G), and $$R_{\mathrm{MB}}(G)$$ (resp. $$S_{\mathrm{MB}}(G)$$ ) denotes the minimum number of moves of Resolver (resp. Spoiler) to win when Resolver has the first move. The corresponding invariants for the game when Spoiler has the first move are denoted by $$R'_{\mathrm{MB}}(G)$$ and $$S'_\mathrm{MB}(G)$$ . Invariants $$R_{\mathrm{MB}}(G)$$ , $$R'_{\mathrm{MB}}(G)$$ , $$S_\mathrm{MB}(G)$$ , and $$S'_{\mathrm{MB}}(G)$$ are compared among themselves and with the metric dimension $$\mathrm{dim}(G)$$ . A large class of graphs G is constructed for which $$R_{\mathrm{MB}}(G) > \mathrm{dim}(G)$$ holds. The effect of twin equivalence classes and pairing resolving sets on the Maker–Breaker resolving game is described. As an application, o(G), as well as $$R_{\mathrm{MB}}(G)$$ and $$R'_{\mathrm{MB}}(G)$$ (or $$S_{\mathrm{MB}}(G)$$ and $$S'_{\mathrm{MB}}(G)$$ ), is determined for several graph classes, including trees, complete multi-partite graphs, grid graphs, and torus grid graphs.

The Balaban index and sum-Balaban index of a connected (molecular) graph G are defined as $$\begin{aligned} J(G)&=\frac{m}{\mu +1} \sum _{uv\in E(G)}\frac{1}{\sqrt{\sigma _{G}(u)\sigma _{G}(v)}}~ \text{ and }\\ SJ(G)&=\frac{m}{\mu +1} \sum _{uv\in E(G)}\frac{1}{\sqrt{\sigma _{G}(u)+\sigma _{G}(v)}}, \end{aligned}$$ respectively, where m is the number of edges, $$\mu $$ is the cyclomatic number, $$\sigma _G(u)$$ is the sum of distances between vertex u and all other vertices of G. In this paper, we establish that $$\begin{aligned} K\left( DS(n-3,\,1)\right)>K\left( DS(n-4,\,2)\right)>\cdots >K \left( DS\left( \left\lceil \frac{n}{2}\right\rceil -1,\, \left\lfloor \frac{n}{2}\right\rfloor -1\right) \right) \end{aligned}$$ $$(K=J,\,SJ),$$ where $$DS(p,\,q)$$ is a double star on $$n\,(=p+q+2,\,p\ge q)$$ vertices. As an application, we determine the extremal graphs of the Balaban index and the sum-Balaban index in the class of chain graphs G on n vertices, where G is a tree or a unicyclic graph. Finally, we give an open problem on Balaban (sum-Balaban) index of connected chain graphs.

In this paper, we consider simple connected weighted graphs in which the edge weights are positive numbers. We obtain several upper bounds on the spectral radius and the signless Laplacian spectral radius of irregular weighted graphs, which extend some known results for irregular unweighted graphs.

An ideal on $$\mathbb N$$ is a family of subsets of $$\mathbb N$$ closed under the operations of taking finite unions and subsets of its elements. The $$\mathcal {I}$$ -open sets of topological spaces, which are determined by an ideal $$\mathcal {I}$$ on $$\mathbb N$$ and the topology of the spaces, are a basic concept of ideal topological spaces. However, it encounters some difficulties in the study of certain structures and mappings of topological spaces. In this paper, we discuss some properties of ideal topological spaces based on $$\mathcal {I}_{sn}$$ -open sets, study the problem generating new topological spaces from ideals, characterize the mappings preserving $$\mathcal {I}$$ -convergence and structure special $$\mathcal {I}$$ -quotient spaces. The following main results are obtained. These show the unique role of $$\mathcal {I}$$ -neighborhood spaces in the study of ideal topological spaces and present a version using the notion of ideals.

Let $k$ be an positive integer. For any two integers $i$ and $j$ in $\{0,1,\dots,k-1\}$, let $|i-j|_k=\min\{|i-j|,k-|i-j|\}$ be the circular distance between $i$ and $j$. Let $t$ be a nonnegative integer. Suppose $f$ is a mapping from $V(G)$ to $\{0,1,\dots,k-1\}$. If adjacent vertices receive different integers, and for each vertex $u$ of $G$, the number of neighbors $v$ of $u$ with $|f(u)-f(v)|_k=1$ is at most $t$, then $f$ is called a $t$-relaxed 2-distant circular $k$-coloring, or simply a $(\frac{k}{2},t)^*$-coloring of $G$. If $G$ has a $(\frac{k}{2},t)^*$-coloring, then $G$ is called $(\frac{k}{2},t)^*$-colorable. In this paper, we prove that, for any two fixed integers $k$ and $t$ with $k\geq2$ and $t\geq1$, deciding whether $G$ is $(\frac{k}{2},t)^*$-colorable is NP-complete expect the case $k=2$ and the case $k=3$ and $t\leq3$, which are polynomially solvable. For any outerplanar graph $G$, e show that all outerplanar graphs are $(\frac{5}{2},4)^*$-colorable, we prove that there is no fixed positive integer $t$ such that all outerplanar graphs are $(\frac{4}{2},t)^*$-colorable., Comment: 19 pages, 8 figures