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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, 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.

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 consider the interval oscillation criteria for second-order damped differential equations with mixed nonlinearities $$\begin{aligned} \left( r(t)(x'(t))^\gamma \right) '+p(t)(x'(t))^\gamma +\sum ^n_{i=0}q_i(t)\left| x(g_i(t))\right| ^{\alpha _i}\text {sgn}\ x(g_i(t))=e(t), \end{aligned}$$ where $$\gamma $$ is a quotient of odd positive integers, $$\alpha _0=\gamma , \alpha _i>0, i=1,\ 2,\ldots ,n$$ with $$r,\ p,\ e$$ , and $$q_i\in C([t_0,\infty ),\mathbb {R}), r(t)>0, g_i:\ \mathbb {R}\rightarrow \mathbb {R}$$ are nondecreasing continuous functions on $$\mathbb {R}$$ and $$\lim _{t\rightarrow \infty }g_i(t)=\infty , i=0,\ 1,\ 2,\ldots ,n.$$ Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

Let G be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of G is such a path in which no two edges have the same color. Let the color degree of a vertex v to be the number of different colors that are used on edges incident to v, and denote it by $$d^c(v)$$ . In a previous paper, we showed that if $$d^c(v)\ge k$$ (color degree condition) for every vertex v of G, then G has a rainbow path of length at least $$\lceil (k+1)/2\rceil $$ . Later, in another paper we first showed that if $$k\le 7$$ , G has a rainbow path of length at least $$k-1$$ , and then, based on this we used induction on k and showed that if $$k\ge 8$$ , then G has a rainbow path of length at least $$\lceil (3k)/5\rceil +1$$ . In 2010, Gyarfas and Mhalla showed that in any proper edge-colored complete graph $$K_n$$ , there is a rainbow path with no less than $$(2n+1)/3$$ vertices. In the present paper, by using a simpler approach we further improve the result by showing that if $$k\ge 8$$ , G has a rainbow path of length at least $$\lceil (2k)/3\rceil +1$$ .

For \(\alpha , \gamma \ge 0\) and \(\beta 0, \, \, \, z\in E \end{aligned}$$ for some \(\phi \in {\mathbb {R}}\). For \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\), we consider the integral transform $$\begin{aligned} V_{\lambda }(f)(z):=\int _{0}^{1}\lambda (t)\frac{f(tz)}{t}\mathrm{d}t, \end{aligned}$$ where \(\lambda \) is a non-negative real-valued integrable function satisfying the condition \(\int _{0}^{1}\lambda (t)\mathrm{d}t=1\). In a very recent paper, Ali et al. (J Math Anal Appl 385:808–822, 2012) discussed the starlikeness of the integral transform \(V_{\lambda }(f)\) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). The aim of present paper is to find conditions on \(\lambda (t)\) such that \(V_{\lambda }(f)\) is starlike of order \(\delta \) (\(0\le \delta \le 1/2\)) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). As applications, we study various choices of \(\lambda (t)\), related to classical integral transforms.

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 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

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.

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

A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set $$[k]=\{1,2,\ldots ,k\}$$ . Let $$\Sigma (u)$$ denote the sum of the color on a vertex u and the colors on all the edges incident with u. For each edge $$uv\in E(G)$$ , if $$\Sigma (u)\ne \Sigma (v)$$ , then we say the coloring distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing total coloring of G. By $$tndi_{\Sigma }(G)$$ , we denote the minimal value of k in such a coloring of G. It has been conjectured by Pilśniak et al. that $$\Delta (G)+3$$ colors enable the existence of a neighbor sum distinguishing total coloring. In this paper, we consider the neighbor sum distinguishing total coloring of corona product $$G\circ H$$ and obtain that $$tndi_{\Sigma }(G\circ H)\le \Delta (G\circ H)+3$$ .

Let $$k\ge 4$$ be an integer, and let G be a graph with maximum degree $$\Delta $$ . In 1991, Alon, McDiarmid and Reed proved that G has a proper coloring using $$O(\Delta ^{(k-1)/(k-2)})$$ colors such that G does not have bichromatic paths with k vertices. In this paper, we improve this result by showing G has a proper coloring using $$(1+\lceil k/2\rceil ^{1/(k-3)})\Delta ^{(k-1)/(k-2)}+\Delta +1$$ colors such that G does not have bichromatic paths with k vertices. We remark that there exists a graph G with maximum degree $$\Delta $$ such that for any proper coloring of G using $$\Omega (\frac{\Delta ^{(k-1)/(k-2)}}{(\log \Delta )^{1/(k-2)}})$$ colors, there is always a bichromatic path with k vertices. Using the similar method, we also show that G has a proper coloring using $$O(\Delta ^{4/3})$$ colors such that G contains neither bichromatic cycles nor bichromatic paths with order 5.

Let A and B be Banach algebras with $$\sigma (B)\ne \emptyset $$ . Let $$\theta , \phi , \gamma \in \sigma (B)$$ and $$\mathrm{Der}(A\times _\theta ^{\phi , \gamma }B)$$ be the set of all linear mappings $$d: A\times B\rightarrow A\times B$$ satisfying $$d((a, b)\cdot _\theta (x, y))=d(a, b)\cdot _\phi (x, y)+ (a, b)\cdot _\gamma d(x, y)$$ for all $$a, x\in A$$ and $$b, y\in B$$ . In this paper, we characterize elements of $$\mathrm{Der}(A\times _\theta ^{\phi , \gamma }B)$$ in the case where A has a right identity. We then investigate the concept of centralizing for elements of $$\mathrm{Der}(A\times _\theta ^{\phi , \gamma }B)$$ and determine dependent elements of $$\mathrm{Der}(A\times _\theta ^{\phi , \gamma }B)$$ . We also apply some results to group algebras.

Let p be an odd prime and $$H_k^{(n)}=\sum _{j=1}^k1/j^n$$ denote the generalized harmonic number. In this paper, the authors establish a kind of congruences involving $$\sum _{k=1}^{p-1}k^mH_{k}^{(n)}\pmod {p^2}$$ , where m, n are positive integers. Furthermore, the authors prove a congruence for $$\sum _{k=1}^{p-1}k^{p-2}(H_{k}^{(2)})^2\pmod {p^2}$$ .

In this paper, we study the existence of multiple solutions for the following biharmonic problem $$\begin{aligned} \Delta ^2 u= & {} f(x,u) + g(x,u)\quad \hbox {in}\quad \Omega ,\\ u= & {} \Delta u =0 \quad \hbox {on}\quad \partial \Omega , \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N, (N > 4)$$ is a smooth bounded domain and $$f(x,\xi )$$ is odd in $$\xi , g(x,\xi )$$ is a perturbation term. By using the variant of Rabinowitz’s perturbation method, under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem.

In this paper, we introduced the class $$\mathcal {S}_{T}^{*}(\lambda )$$ of analytic functions which is related to starlike functions and generating function of generalized telephone numbers. By using bounds on some coefficient functionals for the family of functions with positive real part, we obtain for functions in the class $$\mathcal {S}_{T}^{*}(\lambda )$$ several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants.

In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces $$\mathbb {S}_r$$ S r arising by blowing-up of $$\mathbb {P}^2$$ P 2 at r general points, with $$ r \in \{1, \dots , 8 \}$$ r ∈ { 1 , ⋯ , 8 } . Basic questions when studying the problem of points fattening on an arbitrary variety are what is the minimal growth of the initial sequence and how are the sets on which this minimal growth happens characterized geometrically. We provide a complete answer for del Pezzo surfaces.

For a simple connected graph G, the Q-generating function of the numbers $$N_k$$ of semi-edge walks of length k in G is defined by $$W_Q(t)=\sum \nolimits _{k = 0}^\infty {N_k t^k }$$ . This paper reveals that the Q-generating function $$W_Q(t)$$ may be expressed in terms of the Q-polynomials of the graph G and its complement $$\overline{G}$$ . Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function $$W_Q(t)$$ , we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix $$(\lambda I_n-Q(G))^{-1}$$ . This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs.

Let A and B be rings, U a (B, A)-bimodule and $$T=\begin{pmatrix}A&{}0\\ U&{}B\end{pmatrix}$$ a triangular matrix ring. In this paper, we firstly construct a right recollement of stable categories of Ding injective B-modules, Ding injective T-modules and Ding injective A-modules and then establish a recollement of stable categories of these modules.

Let $$n >3$$ and $$ 0< k < \frac{n}{2} $$ be integers. In this paper, we investigate some algebraic properties of the line graph of the graph $$ {Q_n}(k,k+1) $$ where $$ {Q_n}(k,k+1) $$ is the subgraph of the hypercube $$Q_n$$ which is induced by the set of vertices of weights k and $$k+1$$ . The graph $$ {Q_n}(k,k+1) $$ has a close relation to Johnson graph $$J(n+1,k+1)$$ . In fact, it is the square root of the graph $$J(n+1,k+1)$$ . We will see that when $$ n\ne 2k+1$$ , then the graph $$ {Q_n}(k,k+1) $$ is a non-regular edge-transitive graph; hence, its line graph is a vertex-transitive graph. In the first step, we determine the automorphism groups of these graphs for all values of n, k. In the second step, we study Cayley properties of the line graphs of these graphs. In particular, we show that if $$k\ge 3$$ and $$ n \ne 2k+1$$ , then except for the cases $$(k,n) \ne (3,9)$$ and $$(k,n) \ne (3,33)$$ , the line graph of the graph $$ {Q_n}(k,k+1) $$ is a vertex-transitive non-Cayley graph. Also, we show that the line graph of the graph $$ {Q_n}(1,2) $$ is a Cayley graph if and only if n is a power of a prime p. Moreover, we show that for ‘almost all’ even values of k, the line graph of the graph $$ {Q_{2k+1}}(k,k+1) $$ is a vertex-transitive non-Cayley graph.

In this paper, we consider the nonlinear eigenvalue problems $$\begin{aligned} \begin{aligned}&u''''=\lambda h(t)f(u), \quad 00$$ for $$s\ne 0$$ , and $$f_0=f_\infty =0$$ , $$f_0=\lim _{|s|\rightarrow 0}f(s)/s, \; f_\infty =\lim _{|s|\rightarrow \infty }f(s)/s$$ . We investigate the global structure of one-sign solutions by using bifurcation techniques.

A graph G is $$(d_{1},d_{2},\ldots ,d_{k})$$ -colorable if the vertex set of G can be partitioned into subsets $$V_{1}, V_{2},\ldots , V_{k}$$ such that the subgraph $$G[V_{i}]$$ induced by $$V_{i}$$ has maximum degree at most $$d_{i}$$ for $$i = 1, 2, \ldots , k$$ . Novosibirsk’s Conjecture (Sib lektron Mat Izv 3:428–440, 2006) says that every planar graph without 3-cycles adjacent to cycles of length 3 or 5 is 3-colorable. Borodin et al. (Discrete Math 310: 167–173, 2010) asked whether every planar graph without adjacent cycles of length at most 5 is 3-colorable. Cohen-Addad et al. (J Comb Theory Ser B 122:452–456, 2017) gave a negative answer to both Novosibirsk’s conjecture and Borodin et al.’s problem. Zhang et al. (Discrete Math 339:3032–3042, 2016) asked whether every planar graph without adjacent cycles of length at most 5 is (1, 0, 0)-colorable. In this paper, we show that every planar graph without adjacent cycles of length at most five is (2, 0, 0)-colorable, which improves the result of Chen et al. (Discrete Math 339:886–905, 2016) who proved that every planar graph without cycles of length 4 and 5 is (2, 0, 0)-colorable.

Let G be a nontrivial connected graph with a vertex-coloring c: $$V(G)\rightarrow \{1,2,\ldots ,q\},q\in N$$ . For a set $$S\subseteq V(G)$$ and $$|S|\ge 2$$ , a subtree T of G satisfying $$S\subseteq V(T)$$ is said to be an S-Steiner tree or simply S-tree. The S-tree T is called a vertex-rainbow S-tree if the vertices of $$V(T)\setminus S$$ have distinct colors. Let k be a fixed integer with $$2\le k\le |V(G)|$$ , if every k-subset S of V(G) has a vertex-rainbow S-tree, then G is said to be vertex-rainbow k-tree connected. The k-vertex-rainbow index of G, denoted by $$rvx_{k}(G)$$ , is the minimum number of colors that are needed in order to make G vertex-rainbow k-tree connected. In this paper, we study the 3-vertex-rainbow index of unicyclic graphs and complementary graphs, respectively.

In this paper, we prove that $$cmo^{p}(\mathbb {R}^{n})$$ and $$\Lambda _{n(\frac{1}{p}-1)}$$ , the dual spaces of local Hardy space $$h^{p}(\mathbb {R}^{n})$$ , are coincide with equivalent norms for $$\frac{n}{n+1}

A finite group $G$ is called $\psi$-divisible if $\psi(H)|\psi(G)$ for any subgroup $H$ of $G$, where $\psi(H)$ and $\psi(G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by classifying the finite groups whose all subgroups are $\psi$-divisible. Since the existence of $\psi$-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $\psi$-divisibility property of ZM-groups. In the end, we introduce the concept of $\psi$-normal divisible group, i.e. a group for which the $\psi$-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $\psi$-normal divisible groups which are neither simple nor nilpotent., Comment: 10 pages

For a connected graph G, the Mostar index Mo(G) and the irregularity irr(G) are defined as $$Mo(G)=\sum _{uv\in E(G)}|n_u-n_v|$$ and $$ irr (G)=\sum _{uv\in E(G)}|d_u-d_v|$$ , respectively, where $$d_u$$ is the degree of the vertex u of G and $$n_u$$ denotes the number of vertices of G which are closer to u than to v for an edge uv. In this paper, we focus on the difference $$\Delta M(G)=Mo(G)- irr (G)$$ of graphs G. For trees T of order n, we characterize the minimum and second minimum $$\Delta M(T)$$ of T and the minimum $$\Delta M(Tr(T))$$ of the triangulation graphs Tr(T). The parity of $$\Delta M$$ of cactus graphs is also reported. The effect on $$\Delta M$$ is studied for two local operations of subdivision and contraction of an edge in a tree. A formula for $$\Delta M(S(T))$$ of the subdivision trees S(T) and the upper and lower bounds on $$\Delta M(S(T))- \Delta M(T)$$ are determined with the corresponding extremal trees T. Moreover, three related open problems are proposed to $$\Delta M$$ of graphs.

In the present paper, we show that $$\vartheta $$ -convergent double series spaces $${\mathcal {CS}}_\vartheta $$ and the series space $$\mathcal {BV}$$ of double sequences of bounded variation are BDK-spaces and investigate their $$AK(\vartheta )$$ -space property, where $$\vartheta \in \{p,bp,r\}$$ .

In this paper, we study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc, or a circle, or a "lollipop", or an interval. As an application, we discover a sufficient and necessary condition for the universal real-rootedness of the polynomials, subject to certain sign condition on the coefficients of the recurrence. Moreover, we obtain the sharp bound for all the zeros when they are real., Comment: 14 pages, 1 figure

A Roman dominating function on a graph G is a function $$f:V(G)\rightarrow \{0,1,2\}$$ satisfying the condition that every vertex u for which $$f(u) = 0$$ is adjacent to at least one vertex v for which $$f(v) =2$$ . A Roman dominating function f is called a restrained Roman dominating function if the induced subgraph of G by the vertices with label 0 has no isolated vertex. The weight of a restrained Roman dominating function is the value $$\omega (f)=\sum _{u\in V(G)} f(u)$$ . The minimum weight of a restrained Roman dominating function of G is called the restrained Roman domination number of G and denoted by $$\gamma _{rR}(G)$$ . In this paper, we show that for any graph G of order $$n\ge 5$$ , $$6\le \gamma _{rR} (G)+\gamma _{rR} ({\overline{G}})\le n+5$$ and characterize all the extremal graphs. In addition, we classify all graphs with large restrained Roman domination number.

In this paper, we study the following Chern–Simons–Schrodinger equation $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\omega u+\lambda \Big (\frac{h^{2}(|x|)}{|x|^{2}}+ \int _{|x|}^{+\infty }\frac{h(s)}{s}u^{2}(s)\hbox {d}s\Big )u=g(u) \quad \text{ in }\ {\mathbb {R}}^{2},\\ \displaystyle u\in H_r^1({\mathbb {R}}^{2}), \end{array}\right. } \end{aligned}$$ where $$\omega ,\lambda >0$$ and $$h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\hbox {d}r$$ . Since the nonlinearity g is asymptotically 5-linear at infinity, there would be a competition between g and the nonlocal term. By constrained minimization arguments and the quantitative deformation lemma, we prove the existence of least energy sign-changing radial solution, which changes sign exactly once. Further, we study the concentration of the least energy sign-changing radial solutions as $$\lambda \rightarrow 0$$ .

Let $$K_n-C_{k}$$ denote the graph obtained from the complete graph $$K_n$$ by deleting k edges of a cycle $$C_k$$ , that is, a nearly complete graph. In this paper, we completely determine the structure of the sandpile group of $$K_{n}-C_{k}$$ for $$3\le k\le n-2$$ .

Let $$p\ge 1, \ell \in \mathbb {N}, \alpha ,\beta >-1$$ and $$\varpi =(\omega _0,\omega _1, \ldots , \omega _{\ell -1})\in \mathbb {R}^{\ell }$$ . Given a suitable function f, we define the discrete–continuous Jacobi–Sobolev norm of f as: $$\begin{aligned} \Vert f \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}:= \left( \sum _{k=0}^{\ell -1} \left| f^{(k)}(\omega _{k})\right| ^{p} + \int _{-1}^{1} \left| f^{(\ell )}(x)\right| ^{p} \mathrm{d}\mu ^{\alpha ,\beta }(x)\right) ^{\frac{1}{p}}, \end{aligned}$$ where $$ \mathrm{d}\mu ^{\alpha ,\beta }(x)=(1-x)^{\alpha } (1+x)^{\beta }\mathrm{d}x$$ . Obviously, $$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle 2}}= \sqrt{\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}}$$ , where $$\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}$$ is the inner product $$\begin{aligned} \langle f,g \rangle _{{\scriptscriptstyle \mathsf {s}}}:= \sum _{k=0}^{\ell -1} f^{(k)}(\omega _{k}) \, g^{(k)}(\omega _{k}) + \int _{-1}^{1} f^{(\ell )}(x) \,g^{(\ell )}(x) \mathrm{d}\mu ^{\alpha ,\beta }(x). \end{aligned}$$ In this paper, we summarize the main advances on the convergence of the Fourier–Sobolev series, in norms of type $$L^p$$ , in the continuous and discrete cases, respectively. Additionally, we study the completeness of the Sobolev space of functions associated with the norm $$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}$$ and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in $$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}$$ norm of the partial sum of the Fourier–Sobolev series of orthogonal polynomials with respect to $$\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}$$ .