Let \text{pod}_{-4}(n)$\text{pod}_{-4}(n)$ denote the number of partition quadruples of n$n$ where the odd parts in each partition are distinct. We find many arithmetic properties of \text{pod}_{-4}(n)$\text{pod}_{-4}(n)$ including the following infinite family of congruences: for any integers {\it\alpha}\geq 1${\it\alpha}\geq 1$ and n\geq 0$n\geq 0$,

\begin{eqnarray}\text{pod}_{-4}\biggl(3^{{\it\alpha}+1}n+\frac{5\cdot 3^{{\it\alpha}}+1}{2}\biggr)\equiv 0~(\text{mod}~9).\end{eqnarray} $$\begin{eqnarray}\text{pod}_{-4}\biggl(3^{{\it\alpha}+1}n+\frac{5\cdot 3^{{\it\alpha}}+1}{2}\biggr)\equiv 0~(\text{mod}~9).\end{eqnarray}$$

\text{pod}_{-4}(n)$\text{pod}_{-4}(n)$

The total irregularity of a simple undirected graph G$G$ is defined as \text{irr}_{t}(G)=\frac{1}{2}\sum _{u,v\in V(G)}|d_{G}(u)-d_{G}(v)|$\text{irr}_{t}(G)=\frac{1}{2}\sum _{u,v\in V(G)}|d_{G}(u)-d_{G}(v)|$, where d_{G}(u)$d_{G}(u)$ denotes the degree of a vertex u\in V(G)$u\in V(G)$. Obviously, \text{irr}_{t}(G)=0$\text{irr}_{t}(G)=0$ if and only if G$G$ is regular. Here, we characterise the nonregular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu et al. [‘The minimal total irregularity of graphs’, Preprint, 2014, arXiv:1404.0931v1 ] about the lower bound on the minimal total irregularity of nonregular connected graphs. We show that the conjectured lower bound of 2n-4$2n-4$ is attained only if nonregular connected graphs of even order are considered, while the sharp lower bound of n-1$n-1$ is attained by graphs of odd order. We also characterise the nonregular graphs with the second and the third smallest total irregularity.

The classical first and second Zagreb indices of a graph G$G$ are defined as M_{1}(G)=\sum _{v\in V(G)}d(v)^{2}$M_{1}(G)=\sum _{v\in V(G)}d(v)^{2}$ and M_{2}(G)=\sum _{e=uv\in E(G)}d(u)d(v),$M_{2}(G)=\sum _{e=uv\in E(G)}d(u)d(v),$ where d(v)$d(v)$ is the degree of the vertex v$v$ of G.$G.$ Recently, Furtula et al. [‘On difference of Zagreb indices’, Discrete Appl. Math.178 (2014), 83–88] studied the difference of M_{1}$M_{1}$ and M_{2},$M_{2},$ and showed that this difference is closely related to the vertex-degree-based invariant RM_{2}(G)=\sum _{e=uv\in E(G)}[d(u)-1][d(v)-1]$RM_{2}(G)=\sum _{e=uv\in E(G)}[d(u)-1][d(v)-1]$, the reduced second Zagreb index. In this paper, we present sharp bounds for the reduced second Zagreb index, given the matching number, independence number and vertex connectivity, and we also completely determine the extremal graphs.

Lladó and Moragas [‘Cycle-magic graphs’, Discrete Math.307 (2007), 2925–2933] showed the cyclic-magic and cyclic-supermagic behaviour of several classes of connected graphs. They discussed cycle-magic labellings of subdivided wheels and friendship graphs, but there are no further results on cycle-magic labellings of other families of subdivided graphs. In this paper, we find cycle-magic labellings for subdivided graphs. We show that if a graph has a cycle-(super)magic labelling, then its uniform subdivided graph also has a cycle-(super)magic labelling. We also discuss some cycle-supermagic labellings for nonuniform subdivided fans and triangular ladders.

We consider the number of spanning trees in circulant graphs of {\it\beta}n${\it\beta}n$ vertices with generators depending linearly on n$n$. The matrix tree theorem gives a closed formula of {\it\beta}n${\it\beta}n$ factors, while we derive a formula of {\it\beta}-1${\it\beta}-1$ factors. We also derive a formula for the number of spanning trees in discrete tori. Finally, we compare the spanning tree entropy of circulant graphs with fixed and nonfixed generators.

In this paper we examine solutions in the Gaussian integers to the Diophantine equation ax^{4}+by^{4}=cz^{2}$ax^{4}+by^{4}=cz^{2}$ for different choices of a,b$a,b$ and c$c$. Elliptic curve methods are used to show that these equations have a finite number of solutions or have no solution.

Let \{{\it\phi}_{j}(z):j\geq 1\}$\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue 1/4+t_{j}^{2}$1/4+t_{j}^{2}$. Let {\it\lambda}_{j}(n)${\it\lambda}_{j}(n)$ be the n$n$th Fourier coefficient of {\it\phi}_{j}${\it\phi}_{j}$ and d_{3}(n)$d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for d_{3}(n)$d_{3}(n)$ and {\it\lambda}_{j}(n)${\it\lambda}_{j}(n)$ is considered, leading to the estimate

\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray} $$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$

t_{j}$t_{j}$

{\it\varepsilon}${\it\varepsilon}$

For two given graphs G_{1}$G_{1}$ and G_{2}$G_{2}$, the Ramsey number R(G_{1},G_{2})$R(G_{1},G_{2})$ is the smallest integer N$N$ such that, for any graph G$G$ of order N$N$, either G$G$ contains G_{1}$G_{1}$ as a subgraph or the complement of G$G$ contains G_{2}$G_{2}$ as a subgraph. A fan F_{l}$F_{l}$ is l$l$ triangles sharing exactly one vertex. In this note, it is shown that R(F_{n},F_{m})=4n+1$R(F_{n},F_{m})=4n+1$ for n\geq \max \{m^{2}-m/2,11m/2-4\}$n\geq \max \{m^{2}-m/2,11m/2-4\}$.

Let g\geq 2$g\geq 2$ be a fixed integer. Let \mathbb{N}$\mathbb{N}$ denote the set of all nonnegative integers and let A$A$ be a subset of \mathbb{N}$\mathbb{N}$. Write r_{2}(A,n)=\sharp \{(a_{1},a_{2})\in A^{2}:a_{1}+a_{2}=n\}.$r_{2}(A,n)=\sharp \{(a_{1},a_{2})\in A^{2}:a_{1}+a_{2}=n\}.$ We construct a thin, strongly minimal, asymptotic g$g$-adic basis A$A$ of order two such that the set of n$n$ with r_{2}(A,n)=2$r_{2}(A,n)=2$ has density one.

Let G$G$ be a finite abelian group and A\subseteq G$A\subseteq G$. For n\in G$n\in G$, denote by r_{A}(n)$r_{A}(n)$ the number of ordered pairs (a_{1},a_{2})\in A^{2}$(a_{1},a_{2})\in A^{2}$ such that a_{1}+a_{2}=n$a_{1}+a_{2}=n$. Among other things, we prove that for any odd number t\geq 3$t\geq 3$, it is not possible to partition G$G$ into t$t$ disjoint sets A_{1},A_{2},\dots ,A_{t}$A_{1},A_{2},\dots ,A_{t}$ with r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$.

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval [0,1)$[0,1)$, viewed as \mathbb{R}/\mathbb{Z}$\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

A subset X$X$ of a finite group G$G$ is a set of pairwise noncommuting elements if xy\neq yx$xy\neq yx$ for all x\neq y\in X$x\neq y\in X$. If |X|\geq |Y|$|X|\geq |Y|$ for any other subset Y$Y$ of pairwise noncommuting elements, then X$X$ is called a maximal subset of pairwise noncommuting elements and the size of such a set is denoted by {\it\omega}(G)${\it\omega}(G)$. In a recent article by Azad et al. [‘Maximal subsets of pairwise noncommuting elements of some finite p$p$-groups’, Bull. Iran. Math. Soc.39(1) (2013), 187–192], the value of {\it\omega}(G)${\it\omega}(G)$ is computed for certain p$p$-groups G$G$. In the present paper, our aim is to generalise these results and find {\it\omega}(G)${\it\omega}(G)$ for some more p$p$-groups of interest.

In this paper, we present a decomposition of the elements of a finite field and illustrate the efficiency of this decomposition in evaluating some specific exponential sums over finite fields. The results can be employed in determining the Walsh spectrum of some Boolean functions.

If the centre of a group G$G$ is trivial, then so is the centre of its automorphism group. We study the structure of the centre of the automorphism group of a group G$G$ when the centre of G$G$ is a cyclic group. In particular, it is shown that the exponent of Z(\text{Aut}(G))$Z(\text{Aut}(G))$ is less than or equal to the exponent of Z(G)$Z(G)$ in this case.

Let X$X$ be a simple, connected, p$p$-valent, G$G$-arc-transitive graph, where the subgroup G\leq \text{Aut}(X)$G\leq \text{Aut}(X)$ is solvable and p\geq 3$p\geq 3$ is a prime. We prove that X$X$ is a regular cover over one of the three possible types of graphs with semi-edges. This enables short proofs of the facts that G$G$ is at most 3-arc-transitive on X$X$ and that its edge kernel is trivial. For pentavalent graphs, two further applications are given: all G$G$-basic pentavalent graphs admitting a solvable arc-transitive group are constructed and an example of a non-Cayley graph of this kind is presented.

The best possible constant in a classical inequality due to Bonsall is established by relating that inequality to Young’s. Further, this extends the range of Bonsall’s inequality and yields a reverse inequality. It also provides a better constant in an inequality of Hardy, Littlewood and Pólya.

In this paper, we find a power series expansion of the invariant differential {\it\omega}_{E}${\it\omega}_{E}$ of an elliptic curve E$E$ defined over \mathbb{Q}$\mathbb{Q}$, where E$E$ is described by certain families of Weierstrass equations. In addition, we derive several congruence relations satisfied by the trace of the Frobenius endomorphism of E$E$.

Let G$G$ be a finitely generated group acting on a compact Hausdorff space {\mathcal{X}}${\mathcal{X}}$. We give a fixed point characterisation for the action being amenable. As a corollary, we obtain a fixed point characterisation for the exactness of G$G$.

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called \boldsymbol{{\it\alpha}}$\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space {\mathcal{C}}^{k}(I)${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for {\mathcal{C}}^{k}(I)${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.

Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group G$G$ with rank r$r$, the inequality [G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on G$G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an n$n$-capable group which is not (n+1)$(n+1)$-capable for every n\in \mathbf{N}$n\in \mathbf{N}$.