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In 1935, Erdos proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.

Consider an elliptic quadric $$Q^-(3,q)$$ in $$\mathrm{PG}(3,q)$$ . Let $$\mathcal {E}$$ and $$\mathcal {T}$$ denote the set of all lines of $$\mathrm{PG}(3,q)$$ which meet $$Q^-(3,q)$$ in 0 and 1 point, respectively. In this paper, we characterize the minimum size $$(\mathcal {T}\cup \mathcal {E})$$ -blocking sets and give a different proof for the characterization of minimum size $$\mathcal {E}$$ -blocking sets in $$\mathrm{PG}(3,q)$$ which works for all q. We also discuss whether the main results of this paper (Theorems 1.6 and 1.7) can be extended to ovoids in $$\mathrm{PG}(3,q)$$ .

In this paper, we study a class of regular graphs, which is related to the Grassmann graph. This class of graphs is called the doubled Grassman graph. The Grassmann graph is the class of graphs, which is defined similar to the Johnson graph. This time, however, we are concerned with the subspaces of a vector space, rather than with the subsets of a set. The doubled Grassmann graph is constructed from the Grassmann graph. This graph, was discovered by Biggs and Gardiner. In this paper, we determine the full automorphism group of the doubled Grassmann graph. For determining the automorphism group of the doubled Grassmann graph, we will use the methods used by Mirafzal (Proc. Indian Acad. Sci. (Math Sci.) 129(3) (2019), Art. 34, 8).

Congruences for partitions have received a great deal of attention in literature. Recently, Bringmann et al. (Electron J. Combin. 22(3) (2015) Paper 3.17, 16 pp.) studied overpartitions with restricted odd differences. In this paper, we present a number of Ramanujan-type congruences for these restricted overpartition functions.

The aim of this paper is to state some results on an $$\alpha $$ -nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2) (2015) 271–283), for any fixed automorphism $$\alpha $$ of a group G. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur’s theorem. In the last section of the paper, we study absolute normal subgroups of a finite group.

Let p be a prime, $$\chi $$ denote a Dirichlet character modulo p. For any integer x ( $$1\le x\le p-1$$ ), $${\bar{x}}$$ denotes the integer inverse of x such that $$x{\bar{x}}\equiv 1(\bmod \, p)$$ , we study the following mean value of a kind of character sums with generalized Dirichlet L-functions $$\begin{aligned} {\mathop {\mathop {\sum }\limits _{\chi (-1)=1}}\limits _{\chi \ne \chi _0}} \left| \sum _{x=1}^{p-1}\chi (x+{\bar{x}})\right| ^2|L(1,\chi ,a)|^2, \end{aligned}$$ where $$\chi _0$$ is the principal character modulo p, and $$L(1,\chi ,a)$$ is the generalized Dirichlet L-functions. In this paper, we will use the analytic method and get a sharp asymptotic formula.

Let N be a sufficiently large integer. In this paper, it is proved that, with at most $$O(N^{4/27+\varepsilon })$$ exceptions, all even positive integers up to N can be represented in the form $$p_1^2+p_2^2+p_3^3+p_4^3+p_5^6+p_6^6$$ , where $$p_1,p_2,p_3,p_4,p_5,p_6$$ are prime numbers. This gives a large improvement of a recent result $$O(N^{127/288+\varepsilon })$$ due to Liu (Proc. Indian Acad. Sci. (Math. Sci.) 130(1) (2020) Article ID. 8).

Alspach et al. (J. Austral. Math. Soc. 56(3) (1994) 391–402) constructed an infinite family of tetravalent graphs M(a; m, n) and proved that if $$n\ge 9$$ be odd and $$a^3\equiv 1 (\mathrm{mod}~n)$$ , then M(a; 3, n) is half-arc-transitive. In this paper, we show that if $$a^3\equiv 1 (\mathrm{mod}~n)$$ , then M(a; 3, n) is an infinite family of tetravalent half-arc-transitive Cayley graphs for all integers n except 7 and 14.

Let $$R=K[x_1,\ldots ,x_n]$$ be the polynomial ring in n variables over a field K and I be a monomial ideal generated in degree d. Bandari and Herzog (Eur. J. Combin. 34 (2013) 752–763) conjectured that a monomial ideal I is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper, we give an affirmative answer to the conjecture in the following cases: (i) $${\text {height}}(I)=n-1$$ ; (ii) I contains at least $$n-3$$ pure powers of the variables $$x_1^d,\ldots ,x_{n-3}^d$$ ; (iii) I is a monomial ideal in at most four variables.

In this paper, we establish a non-asymptotic lower bound for the running maximum of an asymptotically almost negatively associated (AANA) family of random variables. The obtained result is applied to characterize the Kolmogorov–Feller weak law of large numbers for these sequences. We also provide a simplified proof of the necessity part in the Baum–Katz law of large numbers in the AANA setting.

An n-color composition is one in which a part of size m can come in m colors (denoted by subscripts). Compositions that read the same when read forwards or backwards are said to be palindromic. In this paper, we study the number of n-color palindromic compositions whose parts have subscripts belonging to a particular arithmetic progression. That is, the subscripts are of the form $$\ell a + b$$ , where $$\ell $$ and b are fixed positive integers and $$a\ge 0$$ is arbitrary. Among our results, we derive an explicit formula for the generating function and provide a connection with Riordan arrays. Finally, we describe bijections between certain restricted classes of palindromic n-color compositions and subsets of ordinary compositions and ternary words.

Let $$G=(V,E)$$ be a graph with the vertex-set V and the edge-set E. Let N(v) denote the set of neighbors of the vertex v of G. The graph G is called irreducible whenever for every $$v,w \in V$$ if $$v \ne w$$ , then $$N(v)\ne N(w).$$ In this paper, we present a method for finding automorphism groups of connected bipartite irreducible graphs. Then, by our method, we determine automorphism groups of some classes of connected bipartite irreducible graphs, including a class of graphs which are derived from Grassmann graphs. Let $$a_0$$ be a fixed positive integer. We show that if G is a connected non-bipartite irreducible graph such that $$c(v,w)=|N(v)\cap N(w)|=a_0$$ when v, w are adjacent, whereas $$c(v,w) \ne a_0$$ , when v, w are not adjacent, then G is a stable graph, that is, the automorphism group of the bipartite double cover of G is isomorphic with the group $${\mathrm{Aut}}(G) \times {\mathbb {Z}}_2$$ . Finally, we show that the Johnson graph J(n, k) is a stable graph.

Let $$G=(V,E)$$ be a graph. A subset $$S\subseteq V$$ is a 2-dominating set of G if each vertex in $$V-S$$ is adjacent to at least two vertices in S. The 2-domination number of G is the cardinality of the smallest 2-dominating set of G. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs $$P(5k+1,3)$$ , $$P(5k+2,3)$$ and $$P(5k+3,3)$$ is $$4k+2$$ , $$4k+3$$ and $$4k+4$$ , respectively. This proves one conjecture due to Bakhshesh et al. (Proc. Indian Acad. Sci. (Math. Sci.) 128 (2018) 17).

To every finite simple graph G, we associate the so-called edge ideal I(G), which is a square-free quadratic monomial ideal generated by the monomials corresponding to the edges of G. In this paper, we determine all the initial graded Betti numbers of edge ideals of triangular graphs and alternate triangular graphs in terms of the underlying graph. We also compute the regularity of edge ideals of triangular and alternate triangular graphs.

Kaplansky (Ann. Math. 46(3) (1945) 474–479) introduced the notion of completely mixed games. Fifty years later in 1995, he wrote another beautiful paper where he gave a set of necessary and sufficient conditions for a skew symmetric matrix game to be completely mixed. In this work, we attempt to answer when bimatrix games will be completely mixed. In particular, we give a set of necessary and sufficient condition for a bimatrix game (A, B) to be completely mixed when A and B are skew symmetric matrices of order 3. We give several examples to show the sharpness of our result. We also formulate a conjecture when A and B are skew symmetric matrices of order 5.

Given a pair of smoothly bounded domains $$D_1, D_2 \subset \mathbb {C}$$ , the purpose of this paper is to obtain an inequality that relates the Caratheodory metrics on $$D_1, D_2, D_1 \cap D_2$$ and $$D_1 \cup D_2$$ .

In this paper, we answer questions raised by Bardakov and Gongopadhyay (Commun. Algebra 43(11) (2015) 4809–4824). We prove that the palindromic width of HNN extension of a group by proper associated subgroups is infinite. We also prove that the palindromic width of the amalgamated free product of two groups via a proper subgroup is infinite (except when the amalgamated subgroup has index two in each of the factors). Combining these results it follows that the palindromic width of the fundamental group of a graph of groups is mostly infinite.

Let R be a ring. The prime intersection graph of ideals of R, denoted by $$G_{P}(R)$$ , is the graph whose vertex set is the collection of all non-trivial (left) ideals of R with two distinct vertices I and J are adjacent if and only if $$I\cap J\ne 0$$ and either one of I or J is a prime ideal of R. We discuss connectedness in $$G_{P}(R)$$ . The concepts of bipartition, planarity and colorability are interpreted. Finally, we introduce the idea of traversability in $$G_{P}({\mathbb {Z}}_n)$$ . The core part of this paper is observed in the ring $${\mathbb {Z}}_n$$ .

An edge-colored graph G is rainbow k-connected, if for every two vertices of G, there are k internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow k-connected coloring of G, $$rc_k(G)$$ , is the rainbow k-connection number of G. Let G and H be two connected graphs, where O is an orientation of G. Let $${\vec {e}}$$ be an oriented edge of H. The edge-comb product of G (under the orientation O) and H on $$\vec {e}$$ , $$G{}^o\rhd _{\vec {e}}H$$ , is a graph obtained by taking one copy of G and |E(G)| copies of H and identifying the i-th copy of H at the edge $${\vec {e}}$$ to the i-th edge of G, where the two edges have the same orientation. In this paper, we provide sharp lower and upper bounds for rainbow 2-connection numbers of edge-comb product of a cycle and a Hamiltonian graph. We also determine the rainbow 2-connection numbers of edge-comb product of a cycle with some graphs, i.e. complete graph, fan graph, cycle graph, and wheel graph.

Let R(n) denote the number of representations of an even integer n as the sum of two squares, two cubes and two sixth powers of primes, and by $$\mathcal {E}(N)$$ we denote the number of even integers $$n \leqslant N$$ such that the expected asymptotic formula for R(n) fails to hold. In this paper, it is proved that $$\mathcal {E}(N) \ll N^{\frac{127}{288} + \varepsilon }$$ for any $$\varepsilon >0$$ .

A composite number is a positive integer that has at least one divisor integer other than 1 and itself. In this paper, we give a detailed structural description of a group if it has a unique composite conjugacy class size.

In this paper, we continue investigating the partition dimension for disconnected graphs. We determine the partition dimension for some classes of disconnected graphs G consisting of two components. If $$G=G_1 \cup G_2$$ , then we give the bounds of the partition dimension of G for $$G_1 = P_n$$ or $$G_1=C_n$$ and also for $$pd(G_1)=pd(G_2)$$ .

The notation $$F\rightarrow (G,H)$$ means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The connected size Ramsey number $$\hat{r}_c(G,H)$$ of graphs G and H is the minimum size of a connected graph F satisfying $$F\rightarrow (G,H)$$ . For $$m \ge 2,$$ the graph consisting of m independent edges is called a matching and is denoted by $$mK_2$$ . In 1981, Erdos and Faudree determined the size Ramsey numbers for the pair $$(mK_2, K_{1,t})$$ . They showed that the disconnected graph $$mK_{1,t} \rightarrow (mK_2,K_{1,t})$$ for $$ t,m \ge 1$$ . In this paper, we will determine the connected size Ramsey number $$\hat{r}_c(nK_2, K_{1,3})$$ for $$n\ge 2$$ and $$\hat{r}_c(3K_2, C_4)$$ . We also derive an upper bound of the connected size Ramsey number $$\hat{r}_c(nK_2, C_4),$$ for $$n\ge 4$$ .

For any given two graphs G and H, the notation $$F\rightarrow $$ (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if $$F\rightarrow $$ (G, H) but $$F-e\nrightarrow (G,H)$$ , for every $$e \in E(F)$$ . The class of all Ramsey (G, H)-minimal graphs is denoted by $$\mathcal {R}(G,H)$$ . In this paper, we construct some infinite families of trees belonging to $$\mathcal {R}(P_3,P_n)$$ , for $$n=8$$ and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to $$\mathcal {R}(P_3,P_n)$$ , for $$n\ge 10$$ .

A graph is called integral, if all of its eigenvalues are integers. In this paper, we give some results about integral pentavalent Cayley graphs on abelian or dihedral groups.

An r-color composition of a positive integer n is a sequence of positive integers, called parts, summing to n in which each part of size r is assigned one of r possible colors. In this paper, we address the problem of counting the r-color compositions having a prescribed number of rises. Formulas for the relevant generating functions are computed which count the compositions in question according to a certain statistic. Furthermore, we find explicit formulas for the total number of rises within all of the r-color compositions of n having a fixed number of parts. A similar treatment is given for the problem of counting the number of levels and a further generalization in terms of rises of a particular type is discussed.

In this paper, we show that the truncated binomial polynomials defined by $P_{n,k}(x)={\sum }_{j=0}^{k} {n \choose j} x^{j}$ are irreducible for each k≤6 and every n≥k+2. Under the same assumption n≥k+2, we also show that the polynomial P n,k cannot be expressed as a composition P n,k (x) = g(h(x)) with $g \in \mathbb {Q}[x]$ of degree at least 2 and a quadratic polynomial $h \in \mathbb {Q}[x]$ . Finally, we show that for k≥2 and m,n≥k+1 the roots of the polynomial P m,k cannot be obtained from the roots of P n,k , where m≠n, by a linear map.

In this paper, we show that for t > 0, the joint distribution of the past {Wt−s: 0 ≤ s ≤ t} and the future {Wt + s:s ≥ 0} of a d-dimensional standard Brownian motion (Ws), conditioned on {Wt ∈ U}, where U is a bounded open set in ℝd, converges weakly in C[0,∞)×C[0,∞) as t→∞. The limiting distribution is that of a pair of coupled processes Y + B1,Y + B2 where Y,B1,B2 are independent, Y is uniformly distributed on U and B1,B2 are standard d-dimensional Brownian motions. Let σt,dt be respectively, the last entrance time before time t into the set U and the first exit time after t from U. When the boundary of U is regular, we use the continuous mapping theorem to show that the limiting distribution as t → ∞ of the four dimensional vector with components \((W_{\sigma _{t}},t-\sigma _{t},W_{d_{t}},d_{t}-t)\), conditioned on {Wt∈U}, is the same as that of the four dimensional vector whose components are the place and time of first exit from U of the processes Y + B1 and Y + B2 respectively.

Multipermutohedron ideals have rich combinatorial properties. An explicit combinatorial formula for the multigraded Betti numbers of a multipermutohedron ideal and their Alexander duals are known. Also, the dimension of the Artinian quotient of an Alexander dual of a multipermutohedron ideal is the number of generalized parking functions. In this paper, monomial ideals which are certain variants of multipermutohedron ideals are studied. Multigraded Betti numbers of these variant monomial ideals and their Alexander duals are obtained. Further, many interesting combinatorial properties of multipermutohedron ideals are extended to these variant monomial ideals.

In this paper, we find non-negative (n, m, a) integer solutions of the diophantine equation $$F_{n}-F_{m}=3^{a}$$ , where $$F_{n}$$ and $$F_{m}$$ are Fibonacci numbers. For proving our theorem, we use lower bounds in linear forms.

Let G be a connected simply connected nilpotent Lie group with discrete uniform subgroup $$\Gamma $$ . A connected closed subgroup H of G is called $$\Gamma $$ -rational if $$H\cap \Gamma $$ is a discrete uniform subgroup of H. For a closed connected subgroup H of G, let $$\mathcal {I}(H, \Gamma )$$ denote the identity component of the closure of the subgroup generated by H and $$\Gamma $$ . In this paper, we prove that $$\mathcal {I}(H, \Gamma )$$ is the smallest normal $$\Gamma $$ -rational connected closed subgroup containing H. As an immediate consequence, we obtain that $$\mathcal {I}(H, \Gamma )$$ depends only on the commensurability class of $$\Gamma $$ . As applications, we give two results. In the first, we determine explicitly the smallest $$\Gamma $$ -rational connected closed subgroup containing H. The second is a characterization of ergodicity of nilflow $$ (G/\Gamma , H)$$ in terms of $$\mathcal {I}(H, \Gamma )$$ . Furthermore, a characterization of the irreducible unitary representations of G for which the restriction to $$\Gamma $$ remain irreducible is given.

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. In this paper, we classify connected heptavalent symmetric graphs of order 24p for each prime p. As a result, there are twelve sporadic such graphs: one for $$p=2$$ , four for $$p=3$$ , one for $$p=5$$ and six for $$p=13$$ .

On pages 255–256 of his second notebook, Ramanujan recorded an Eisenstein series identity of level 15 without offering a proof. Previously, Berndt (Ramanujan’s Notebooks: Part III (1991) (New York: Springer)) proved this identity using the theory of modular forms. In this paper, we give an elementary proof of this identity. In the process, we also give an elementary proof of three Ramanujan’s $$P-Q$$ identities of level 15. Further, using the $$P-Q$$ identities we prove four Ramanujan type Eisenstein series of level 15 due to Cooper and Ye (Trans. Am. Math. Soc. 368 (2016) 7883–7910), where they have proved using the theory of modular forms.

Let $$\chi $$ denote the class of all groups G such that $$\Phi (G)\cap Z(G)=1.$$ In this paper, it is shown that the converse of Baer’s theorem holds for the groups in $$\chi .$$ Then we prove that the existence of the isomorphism between the center factors of the groups in $$\chi $$ suffices for those groups to be isoclinic. We also prove that the isoclinism coincides with the n-isoclinism in $$\chi $$ . Finally, we obtain a criterion for c-capability of finite groups in $$\chi $$ .

In this paper, we consider determinants of some families of Toeplitz–Hessenberg matrices having various translates of the Catalan numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Catalan numbers and multinomial coefficients. Combinatorial proofs may be given for several of the identities that are obtained.

Let p be a prime $$\equiv 1 \!\!\!\pmod {11}$$ . If an integer D with $$(p,D)=1$$ is an eleventh power nonresidue $$\pmod {p}$$ , then $$D^{(p-1)/11} \equiv \alpha \, \!\!\!\pmod {p}$$ for some eleventh root of unity $$\alpha (\not \equiv 1)\,\!\!\!\pmod {p}$$ . In this paper, we establish an explicit expression for $$\alpha $$ in terms of a particular solution of certain quadratic partition of p. Euler’s criterion for eleventh power residues and nonresidues is given with explicit results for $$D=\displaystyle {2,7,11}$$ .

In this paper, the open problem posed by Sareen and Rana (Proc. Indian Acad. Sci. (Math. Sci.) 126 (2016) 549–556) is addressed. Here, we interpret two tenth order mock theta functions combinatorially in terms of lattice paths. Then we extend enumeration of one of these with Bender–Knuth matrices; the other by using Frobenius partitions. The combinatorial interpretation of one of these mock theta functions in terms of Frobenius partitions gives an answer to the open problem. Finally, we establish bijections between different classes of combinatorial objects which lead us to one 4-way and one 3-way combinatorial identity.

Let R be a commutative ring with $$1\ne 0$$ and the additive group $$R^+$$ . Several graphs on R have been introduced by many authors, among zero-divisor graph $$\Gamma _1(R)$$ , co-maximal graph $$\Gamma _2(R)$$ , annihilator graph AG(R), total graph $$ T(\Gamma (R))$$ , cozero-divisors graph $$\Gamma _\mathrm{c}(R)$$ , equivalence classes graph $$\Gamma _\mathrm{E}(R)$$ and the Cayley graph $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with $$|\mathrm{Max}(R)|=n \ge 3$$ , $$ \Gamma _1(R) \simeq \Gamma _2(R)$$ if and only if $$R\simeq \mathbb {Z}^n_2$$ ; if and only if $$\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)$$ . Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.

Let A be a nonempty finite subset of an additive abelian group G and let r and h be positive integers. The generalized h-fold sumset of A, denoted by $$h^{(r)}A$$ , is the set of all sums of h elements of A, where each element appears in a sum at most r times. The direct problem for $$h^{(r)}A$$ is to find a lower bound for $$|h^{(r)}A|$$ in terms of |A|. The inverse problem for $$h^{(r)}A$$ is to determine the structure of the finite set A for which $$|h^{(r)}A|$$ is minimal with respect to some fixed value of |A|. If $$G = \mathbb {Z}$$ , the direct and inverse problems are well studied. In case of $$G = \mathbb {Z}/p\mathbb {Z}$$ , p a prime, the direct problem has been studied very recently by Monopoli (J. Number Theory, 157 (2015) 271–279). In this paper, we express the generalized sumset $$h^{(r)}A$$ in terms of the regular and restricted sumsets. As an application of this result, we give a new proof of the theorem of Monopoli and as the second application, we present new proofs of direct and inverse theorems for the case $$G = \mathbb {Z}$$ .

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of $$E_{m,p} : y^2=x^3-m^2x+p^2$$ , where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of $$E_{m,p}(\mathbb {Q})$$ is trivial for both the cases { $$m\ge 1$$ , $$m\not \equiv 0\pmod 3$$ } and { $$m\ge 1$$ , $$m \equiv 0 \pmod 3$$ , with $$gcd(m,p)=1$$ }. We also show that given any odd prime p and for any positive integer m with $$m\not \equiv 0\pmod 3$$ and $$m\equiv 2\pmod {32}$$ , the lower bound for the rank of $$E_{m,p}(\mathbb {Q})$$ is 2. Finally, we find curves of rank 9 in this family.

In this paper, we characterize primes of the form x2 + dy2 with x≡0(mod N) or y≡0(mod N) for positive integers N and d with d being square free.

To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study arrangements of pseudohyperplanes; they correspond to non-realizable oriented matroids. These arrangements arise as a consequence of the Folkman-Lawrence topological representation theorem. We propose a generalization of the complexification process in this context. In particular we construct a space naturally associated with these pseudo-arrangements which is homeomorphic to the complexified complement in the realizable case. Further we generalize the classical theorem of Salvetti and show that this space has the homotopy type of a cell complex defined in terms of the oriented matroid., Comment: 19 pages, 3 figures. Third version: some more typos fixed, minor changes, final version. To appear in Proceedings Mathematical Sciences. In the second version: typos fixed, exposition improved, contact information updated. arXiv admin note: text overlap with arXiv:1110.1520

In this paper, we enumerate the parameter matrices of all perfect 2-colorings of the generalized Petersen graphs GP (n,2), where n ≥ 5. We also present some basic results for GP (n, k), where n ≥ 5 and k ≥ 3.

A subgroup H of a group G is said to be \(\mathcal {M}^{*}\)-supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is \(\mathcal {M}\)-supplemented in G. In this paper, we prove as follows: Let E be a normal subgroup of a group G. Suppose that every maximal subgroup of every non-cyclic Sylow subgroup P of F∗(E) is \(\mathcal {M}^{*}\)-supplemented in G, then \(E\leq Z_{\mathcal {U}{\Phi }}(G)\).

Recently several authors investigated structures of generalized bi-circular projections in spaces where the descriptions of the group of surjective isometries are known. Following the same idea in this paper we give complete descriptions of generalized 3-circular projections for symmetric norms on \(\mathbb {C}^{n}\) and \(\mathbb {M}_{m \times n}(\mathbb {C})\).

Quantitative versions of the central results of the metric theory of continued fractions were given primarily by C. De Vroedt. In this paper we give improvements of the bounds involved . For a real number x, let $$x \ = \ c_{0} + \frac{1}{\displaystyle c_{1} + \frac{1}{\displaystyle c_{2} + \frac{1}{\displaystyle c_{3} + \frac{1}{\displaystyle c_{4} +_{\ddots}}}}}. $$ A sample result we prove is that given 𝜖 > 0, $$\left (c_{1} (x) {\cdots} c_{n}(x) \right )^{\frac{1}{n}} = { \prod}_{k=1}^{\infty}\left (1+ \frac{1}{k(k+2)} \right )^{\frac{\log k }{ \log 2}} + o \left (n^{-\frac{1}{ 2}}(\log n )^{\frac{3}{ 2}} (\log \log n)^{\frac{1}{2}+\epsilon} \right ) $$ almost everywhere with respect to the Lebesgue measure.

A graph G is called (k,d)∗-choosable if for every list assignment L satisfying ∣L(v)∣ ≥k for all v∈V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without intersecting i-cycles for all i=3,4,5 is (3,1)∗-choosable.

In [5], a new combinatorial model with four types of points and nine types of lines of the slim dense near hexagon Q(5,2)⊗Q(5,2) was provided and it was then used to construct a non-abelain representation of Q(5,2)⊗Q(5,2) in the extraspecial 2-group \(2_{-}^{1+18}\). In this paper, we give a direct proof for the existence of a non-abelian representation of Q(5,2)⊗Q(5,2) in \(2_{-}^{1+18}\).

The Wiener-type invariants of a simple connected graph G = (V, E) can be expressed in terms of the quantities $W_{f}=\sum_{\{u,v\}\subseteq V}f(d_{G}(u,v))$ for various choices of the function f(x), where d G (u,v) is the distance between vertices u and v in G. In this paper, we give some sufficient conditions for a connected graph to be Hamiltonian, a connected graph to be traceable, and a connected bipartite graph to be Hamiltonian in terms of the Wiener-type invariants.

In this paper we give a new characterization of simple group L2(p) with p a prime by both its order and nse(L2(p)), the set of numbers of elements of L2(p) with the same order.