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In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space (X,d) is a ring if and only if every subset A ? X has one of the following properties: ? A is Bourbaki-bounded, i.e., every uniformly continuous function on X is bounded on A. ? A contains an infinite uniformly isolated subset, i.e., there exist ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.

This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

In this paper, we apply the Faber polynomial expansions to find upper bounds for the general coefficients |amk+1| (k >= 3) of functions in the subclass N?m (?,?;?). The results presented in this paper would generalize and improve some recent works.

For a Tychonoff space $X$, we denote by $(C(X), \tau_k, \tau_p)$ the bitopological space of all real-valued continuous functions on $X$ where $\tau_k$ is the compact-open topology and $\tau_p$ is the topology of pointwise convergence. In papers [5, 6, 13] variations of selective separability and tightness in $(C(X), \tau_k, \tau_p)$ were investigated. In this paper we continued to study the selective properties and the corresponding topological games in the space $(C(X), \tau_k, \tau_p)$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1805.04363

Zagreb indices and their modified versions of a molecular graph originate from many practical problems such as two dimensional quantitative structure-activity (2D QSAR) and molecular chirality. Nowadays, they have become important invariants which can be used to characterize the properties of graphs from different aspects. LetVk n (or Ek n respectively) be a set of graphs of n vertices with vertex connectivity (or edge connectivity respectively) at most k. In this paper, we explore some properties of the modified first and second multiplicative Zagreb indices of graphs in Vkn and Ekn. By using analytic and combinatorial tools, we obtain some sharp lower and upper bounds for these indices of graphs in Vk n and Ekn. In addition, the corresponding extremal graphs which attain the lower or upper bounds are characterized. Our results enrich outcomes on studying Zagreb indices and the methods developed in this paper may provide some new tools for investigating the values on modified multiplicative Zagreb indices of other classes of graphs.

Let k be a nonzero complex number. In this paper we consider k - circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k=1 , the inverses of such ( invertible ) matrices are obtained (in a way different from the way presented in the paper: M. Bahsi and S. Solak , On the Circulant Matrices with Arithmetic Sequence , Int. J. Contemp . Math. Sci. 5 (25) (2010), 1213-1222, and the Moore-Penrose inverses of such (singular) matrices are derived.

Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$ be two bounded linear operators. The general split common fixed-point problem under consideration in this paper is to $$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$ where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$ and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are quasi-nonexpansive mappings with nonempty common fixed-point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above problem (1) allows asymmetric and partial relations between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common fixed-point problem (1) reduces to the general split common fixed-point problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

Suppose $R$ is an associative ring with identity 1. In this paper, We give the necessary and sufficient conditions for the existence and explicit representations of the group inverses of the following two classes block matrices. (i) $M=\left(\begin{array}{cc}AX+YBA (ii) $M=\left(\begin{array}{cc}A&B\\C&D\end{array}\right)$, where $A^{\sharp}$ exists, $A^{\pi}B=0$ and $(D-CA^{\sharp}B)^{\sharp}$ exists.\\ The paper's results generalize some relative results of Li et al. (J. Harbin Engineering Univ., 34, 658-661, 2013), Cao et al. (Appl. Math. Comput., 217, 10271-10277, 2011), Bu et al. (Appl. Math. Comput., 215, 132-139, 2009), Deng et al. (Linear Algebra Appl, 435, 2766-2783, 2011), and Liu et al. (J. Appl. Math. http://dx.doi.org/10.1155 /2013/247028). Some examples are given to illustrate our results.

The present paper studies several types of compatible adjacency relations for digital products such as a $C$-compatible adjacency (or the $L_C$-property in \cite{H13}), an $S$-compatible adjacency in \cite{H19} (or the $L_S$-property in \cite{H13}), which can contribute to the study of product properties of digital spaces (or digital images). Furthermore, to study an automorphism group of a Cartesian product of two digital coverings which do not satisfy a radius $2$ local isomorphism, which remains open, the paper uses some properties of an ultra regular covering in \cite{H16}. By using this approach, we can substantially classify digital products. In particular, using a $C$-compatible adjacency (or the $L_C$-property), we address a product problem of a digital isomorphism (see Theorems 3.6 and 4.1).

The purpose of this paper is to prove a general fixed point theorem for mappings involving almost altering distances and satisfying a new type of common limit range property which generalize the results from Theorem 2.9 [19]. In the last part of the paper, as applications, some fixed point results for mappings satisfying contractive conditions of integral type for almost contractive mappings for ?-contractive mappings and (?,?) - weak contractive mappings in metric spaces are obtained.

In this paper, the complete moment convergence and the integrability of the supremum for weighted sums of negatively orthant dependent (NOD, in short) random variables are presented. As applications, the complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for NODrandom variables are obtained. The results established in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.

In the present paper, we introduce a new type of Bernstein-Durrmeyer operators preserving linear functions in movable interval. The approximation rate of the new operators for continuous functions and Voronovskaja?s asymptotic estimate are obtained.

Recently Srivastava et al. (26) introduced the incomplete Pochhammer symbols by means of the incomplete gamma functions (s; x) and (s; x), and defined incomplete hypergeometric functions whose a number of interesting and fundamental properties and characteristics have been investigated. Further, Cetinkaya (6) introduced the incomplete second Appell hypergeometric functions and studied many interesting and fundamental properties and characteristics. In this paper, motivated by the above- mentioned works, we introduce two incomplete Srivastava's triple hypergeometric functions H and H by using the incomplete Pochhammer symbols and investigate certain properties, for example, their various integral representations, derivative formula, reduction formula and recurrence relation. Various (known or new) special cases and consequences of the results presented here are also considered. 1. Introduction, Definitions and Preliminaries Throughout this paper, N, Z , and C denote the sets of positive integers, negative integers, complex numbers, respectively, and N0 := N(f0g and Z 0 := Z (f0g. The familiar incomplete Gamma functions (s; x) and (s; x) defined by

The aim of this paper is to continue our investigations in line of our recent paper, Savas [24] and [26]. We introduce the notion of AI-double statistical convergence which includes the new summabilitymethods studied in [24] and [23] as special cases and make certain observations on this new and moregeneral summability method. © 2016, University of Nis. All rights reserved.

Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

In this paper we study the notion of a graded $\Omega$-group $(X,+,\Omega),$ but graded in the sense of M.~Krasner, i.e., we impose nothing on the grading set except that it is nonempty, since operations of $\Omega$ and the grading of $(X,+)$ induce operations (generally partial) on the grading set. We prove that graded $\Omega$-groups in Krasner's sense are determined up to isomorphism by their homogeneous parts, which, with respect to induced operations, represent partial structures called $\Omega$-\emph{homogroupoids}, thus narrowing down the theory of graded $\Omega$-groups to the theory of $\Omega$-homogroupoids. This approach already proved to be useful in questions regarding A. V. Kelarev's $S$-graded rings inducing $S,$ where $S$ is a partial cancellative groupoid. Particularly, in this paper we prove that the homogeneous subring of a Jacobson $S$-graded ring inducing $S$ is Jacobson under certain assumptions. We also discuss the theory of prime radicals for $\Omega$-homogroupoids thus extending results of A.~V.~Mikhalev, I.~N.~Balaba and S.~A.~Pikhtilkov in a natural way. We study some classes of $\Omega$-homogroupoids for which the lower and upper weakly solvable radicals coincide and also, study the question of the homogeneity of the prime radical of a graded ring.

In this paper, we introduce and investigate three interesting subclass $\Sigma_\vartheta^*(\lambda,\alpha)$, $\Sigma_\vartheta^B(\beta,\alpha)$ and $\tilde{\Sigma}_\vartheta^*(\alpha,\beta)$ of meromorphic and bi-univalent functions on $\Delta=\{z\in\mathbb{C}: |z|>1\}$, obtain their coefficient estimates. The results presented in this paper generalize the recent work S.A.Halim et al.[1].

The Tur?n number ex(n,H) of a graph H, is the maximum number of edges in a graph of order n which does not contain H as a subgraph. Let Ex(n,H) denote all H-free graphs on n vertices with ex(n,H) edges. Let Pi denote a path consisting of i vertices, and mPi denote m disjoint copies of Pi. In this paper, we give the Tur?n number ex(n,3P5) for all positive integers n, which partly solve the conjecture proposed by L. Yuan and X. Zhang [7]. Moreover, we characterize all extremal graphs of 3P5 denoted by Ex(n, 3P5).

The purpose of this paper is to determine L(G), the absolute center of the group G, when G is a p-group of maximal class. Particularly we find L(G) for metabelian p-groups of maximal class, all p-groups of maximal class of order less than p6 and p-groups of maximal class for p = 2,3.

The Modular group ? is the most well-known discrete group with many applications. This work investigates some subgraphs of the subgroup ?3, defined by {(ab cd)??:ab+cd ?0 (mod 3)}. In [1], the subgraph F1,1 of the subgroup ?3 ? ? is studied, and Fibonacci numbers are obtained by means of the subgraph of F1,1. In this paper, we give a generalization of the subgraphs generating Fibonacci numbers for the subgroup ?3 and some subgraphs having special conditions.

The aim of this paper is to construct generating functions for Boole-type combinatorial numbers and polynomials. Using these generating functions, we derive not only fundamental properties of these numbers and polynomials, but also some identities and formulas. Finally, we present a brief historical remarks and observations on our generating functions and Peters and Boole-type numbers and polynomials.

The aim of this paper is devoted to present some new results on six types of special mappings between L-convex spaces. For this purpose, we first make a summary about some novel elementary properties of L-CP mappings and L-CC mappings. Secondly, we propose the definitions of almost LCC mappings, L-isomorphic mappings and L-embedding mappings and investigate their fundamental characterizations. Finally, we establish the connections between L-quotient mappings and L-quotient spaces. As a summary, we give a diagram to show the relationships among all the above-mentioned mappings.

Let Zn2 be the elementary abelian 2-group, which can be viewed as the vector space of dimension n over F2. Let {e1,..., en} be the standard basis of Zn2 and ?k = ek +...+ en for some 1 ? k ? n-1. Denote by ?n,k the Cayley graph over Zn2 with generating set Sk = {e1,..., en,?k}, that is, ?n,k = Cay(Zn2,Sk). In this paper, we characterize the automorphism group of ?n,k for 1 ? k ? n-1 and determine all Cayley graphs over Zn2 isomorphic to ?n,k. Furthermore, we prove that for any Cayley graph ? = Cay(Zn2,T), if ? and ?n,k share the same spectrum, then ? ? ?n,k. Note that ?n,1 is known as the so called n-dimensional folded hypercube FQn, and ?n,k is known as the n-dimensional enhanced hypercube Qn,k.

Given a graph G and a positive integer i, an i-packing in G is a subset W of the vertex set of G such that the distance between any two distinct vertices from W is greater than i. The packing chromatic number of a graph G, ??(G), is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ? {1,..., k}, where each Vi is an i-packing. In this paper, we present some general properties of packing chromatic numbers of finite super subdivisions of graphs. We determine the packing chromatic numbers of the finite super subdivisions of complete graphs, cycles and some neighborhood corona graphs.

In this paper, we introduce new concepts of (m,q)-isometries and (m,?)-isometries tuples of commutative mappings on metrics spaces. We discuss the most interesting results concerning this class of mappings obtained form the idea of generalizing the (m,q)-isometries and (m,?)-isometries for single mappings. In particular, we prove that if T = (T1,..., Tn) is an (m,q)-isometric commutative and power bounded tuple, then T is a (1,q)-isometric tuple. Moreover, we show that if T = (T1,...,Td) is an (m,?)- isometric commutative tuple of mappings on a metric space (E,d), then there exists a metric d? on E such that T is a (1,?)-isometric tuple on (E,d?).

In this paper we illustrate a tight interplay between homotopy theory and combinatorics within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related small cover. We finish by studying Massey products in the cohomology ring of moment-angle manifolds over the dodecahedron and how the existence of nontrivial Massey products influences the behaviour of the Poincar? series of the corresponding Pontryagin algebra.

In this paper we provide a detailed study of the Banach space of strongly (p,q)-summable sequences. We prove that this space is a topological dual of a class of mixed (s,p)-summable sequences, showing in this way new properties of this space. We apply these results to obtain the characterization of the adjoints of (r,p,q)-summing operators.

In this paper, we continue the study of the L-Grundy domination number of a graph introduced and first studied in [Grundy dominating sequences and zero forcing sets, Discrete Optim. 26 (2017) 66-77]. A vertex in a graph dominates itself and all vertices adjacent to it, while a vertex totally dominates another vertex if they are adjacent. A sequence of distinct vertices in a graph G is called an L-sequence if every vertex v in the sequence is such that v dominates at least one vertex that is not totally dominated by any vertex that precedes v in the sequence. The maximum length of such a sequence is called the L-Grundy domination number, L gr(G), of G. We show that the L-Grundy domination number of every forest G on n vertices equals n, and we provide a linear-time algorithm to find an L-sequence of length n in G. We prove that the decision problem to determine if the L-Grundy domination number of a split graph G is at least k for a given integer k is NP-complete. We establish a lower bound on L gr(G) when G is a regular graph, and investigate graphs G on n vertices for which L gr(G) = n.

In this paper, we study cyclic codes of length n over R = Zq + uZq, u2 = 0, where q is a power of a prime p and (n; p) = 1. We have determined the complete ideal structure of R. Using this, we have obtained the structure of cyclic codes and that of their duals through the factorization of xn-1 over R. We have also computed total number of cyclic codes of length n over R. A necessary and sufficient condition for a cyclic code over R to be self-dual is presented. We have presented a formula for the total number of self-dual cyclic codes of length n over R. A new Gray map from R to Z2rp is defined. Using Magma, some good cyclic codes of length 4 over Z9 + uZ9 are obtained.

Let $B^p_{\sigma}$, $1\le p0$, denote the space of all $f\in L^p(\mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-\sigma,\sigma]$. The classical sampling theorem states that each $f\in B^p_{\sigma}$ may be reconstructed exactly from its sample values at equispaced sampling points $\{\pi m/\sigma\}_{m\in\mathbb{Z}} $ spaced by $\pi /\sigma$. Reconstruction is also possible from sample values at sampling points $\{\pi \theta m/\sigma\}_m $ with certain $1< \theta\le 2$ if we know $f(\theta\pi m/\sigma) $ and $f'(\theta\pi m/\sigma)$, $m\in\mathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives., Comment: 10 pages

In this paper the solutions of the following difference equation is examined: xn+1 = xn-7/1 + xn-3, n = 0, 1, 2, 3,... where the initial conditions are positive real numbers.

In this paper, we establish some Hermite-Hadamard type inequalities for h- convex function on high-dimensional balls and ellipsoids, which extend some known results. Some mappings connected with these inequalities and related results are also obtained.

© University of Nis. All rights reserved. The aim of this paper is to obtain new lower bounds for the smallest singular value for some special subclasses of nonsingular H-matrices. This is done in two steps: first, unifying principle for deriving new upper bounds for the norm 1 of the inverse of an arbitrary nonsingular H-matrix is presented, and then, it is combined with some well-known upper bounds for the infinity norm of the inverse. The importance and efficiency of the results are illustrated by an example from ecological modelling, as well as on a type of large-scale matrices posessing a block structure, arising in boundary value problems.

In this paper, we study (1 + 2u + 2v)-constacyclic and skew (1 + 2u + 2v)-constacyclic codes over the ring Z4 + uZ4 + vZ4 + uvZ4 where u2 = v2 = 0,uv = vu. We define some new Gray maps and show that the Gray images of (1 + 2u + 2v)-constacyclic and skew (1 + 2u + 2v)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z4. Further, we determine the structure of (1 + 2u + 2v)-constacyclic codes of odd length n.

In this paper we continue the study of signed double Roman dominating functions in graphs. A signed double Roman dominating function (SDRDF) on a graph G = (V,E) is a function f : V(G)  {-1,1,2,3} having the property that for each v  V(G), f [v] ≥ 1, and if f (v)= -1, then vertex v has at least two neighbors assigned 2 under f or one neighbor w with f(w)=3, and if f(v)=1, then vertex v must have at leat one neighbor w with f (w)≥2. The weight of a SDRDF is the sum of its function values over all vertices. The signed double Roman domination number γsdR(G) is the minimum weight of a SDRDF on G. We present several lower bounds on the signed double Roman domination number of a graph in terms of various graph invariants. In particular, we show that if G is a graph of order n and size m with no isolated vertex, then γsdR(G) ≥ 19n-24m/9 and γsdR(G)≥4 √n/3-n. Moreover, we characterize the graphs attaining equality in these two bounds.

We consider the class Ap of functions f analytic in the unit disk |z| < 1 in the complex plane, of the form f(z) = zp + ... such that Rezf(p)(z)/f (p-1)(z) > 0 in the unit disc. The object of the present paper is to derive some bounds for coefficients in this class and relation with the functions satisfying condition Ref(k)(z)/f(p-k)(z) > 0 in the unit disc.

In this paper, by using Orlicz function and almost lacunary bounded variation we introduce and examine a new sequence space in 2- normed spaces. We also study some basic topological and algebraic properties of these spaces. In addition, we shall established inclusion theorems between these sequence spaces. © 2019, University of Nis. All right reserved.

In this paper, we present new relationships between the terms of sequence {Rn} and {Sn} and permanents of some upper Hessenberg matrices.

Let ?1(G) and q1(G) be the spectral radius and the signless Laplacian spectral radius of a kuniform hypergraph G, respectively. In this paper, we give the lower bounds of d-?1(H) and 2d-q1(H), where H is a proper subgraph of a f (-edge)-connected d-regular (linear) k-uniform hypergraph. Meanwhile, we also give the lower bounds of 2?-q1(G) and ?-?1(G), where G is a nonregular f (-edge)-connected (linear) k-uniform hypergraph with maximum degree ?.

The weighted vertex PI index of a graph G is defined by PIw(G)=?e=uv?E(G) (dG(u)+dG(v))(nu(e|G) + nv(e|G)) where dG(u) denotes the vertex degree of u and nu(e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. A graph is a cactus if it is connected and all its blocks are either edges or cycles. In this paper, we give the upper and lower bounds on the weighted vertex PI index of cacti with n vertices and s cycles, and completely characterize the corresponding extremal graphs.

The preliminary idea of statistical weighted B-summability was introduced by Kadak et al. [27]. Subsequently, deferred weighted statistical B-summability has recently been studied by Pradhan et al. [38]. In this paper, we study statistical versions of deferred weighted B-summability as well as deferred weighted B-convergence with respect to the difference sequence of order r (> 0) involving (p,q)-integers and accordingly established an inclusion between them. Moreover, based upon our proposed methods, we prove an approximation theorem (Korovkin-type) for functions of two variables defined on a Banach space CB(D) and demonstrated that, our theorem effectively improves and generalizes most (if not all) of the existing results depending on the choice of (p,q)-integers. Finally, with the help of the modulus of continuity we estimate the rate of convergence for our proposed methods. Also, an illustrative example is provided here by generalized (p,q)-analogue of Bernstein operators of two variables to demonstrate that our theorem is stronger than its traditional and statistical versions.

Petrovic in [M. Petrovic, On graphs with exactly one eigenvalue less than -1, J. Combin. Theory Ser. B 52 (1991) 102-112] determined all connected graphs with exactly one eigenvalue less than -1 and all minimal graphs with exactly two eigenvalues less than -1. By using these minimal graphs, in this paper, we determine all connected graphs having ??1 as an eigenvalue with multiplicity n-4 or n-5.

LetG = (V,E) be a simple connected graph of order n (?2) and size m, where V(G) = {1, 2,..., n}. Also let ? = d1 ? d2 ?... ? dn = ? > 0, di = d(i), be a sequence of its vertex degrees with maximum degree ? and minimum degree ?. The symmetric division deg index, SDD, was defined in [D. Vukicevic, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta 83 (2010) 261- 273] as SDD = SDD(G) = ?i~j d2i+d2j/didj, where i~j means that vertices i and j are adjacent. In this paper we give some new bounds for this topological index. Moreover, we present a relation between topological indices of graph.

In this paper, we give a simple proof for the boundary Schwarz lemma at the upper half plane. Considering that f(z) is a holomorphic function defined on the upper half plane, we derive inequalities for the modulus of derivative of f (z), |f'(0)| by assuming that the f(z) function is also holomorphic at the boundary point z = 0 on the real axis with f(0)=Rf(i).

In this paper, we investigate the weighted B-distances of infinite sequences. The general neighbourhood sequences were introduced for measuring distances in digital geometry (Zn), and the theory was recently extended for application to sequences. By assigning various weights to the elements of the sequences the concept is further generalized. An algorithm is presented which provides a shortest path between two sequences. Formula is also provided to calculate the weighted B-distance of any two sequences with a neighbourhood sequence B and a weight sequence. There are several neighbourhood sequences, which do not generate metrics. We prove a necessary and sufficient condition for a B-distance to define a generalized metric space above the sequences. Moreover, our results can be applied if the elements of the sequences used with various weights. In case of weight functions used B-distances we present also the metric conditions.

The eccentricity of a vertex is the maximal distance from it to another vertex and the average eccentricity ecc(G) of a graph G is the mean value of eccentricities of all vertices of G. A set S ? V(G) is a dominating set of a graph G if NG(v) ? S ? 0 for any vertex v ? V(G)\S. The domination number (G) of G is the minimum cardinality of all dominating sets of G. In this paper, we correct an AutoGraphiX conjecture regarding the domination number and average eccentricity, and present a proof of the revised conjecture. In addition, we establish an upper bound on ?(T)-ecc(T) for an n-vertex tree T.

Let D and D' be two digraphs with the same vertex set V, and let F be a set of positive integers. The digraphs D and D' are hereditarily isomorphic whenever the (induced) subdigraphs D[X] and D'[X] are isomorphic for each nonempty vertex subset X. They are F-isomorphic if the subdigraphs D[X] and D'[X] are isomorphic for each vertex subset X with |X|? F. In this paper, we prove that if D and D' are two {4,n-3}-isomorphic n-vertex digraphs, where n ? 9, then D and D0 are hereditarily isomorphic. As a corollary, we obtain that given integers k and n with 4 ? k ? n-6, if D and D' are two {n-k}-isomorphic n-vertex digraphs, then D and D' are hereditarily isomorphic.

Let I(G) be a topological index of a graph. If I(G + e) < I(G) (or I(G + e) > I(G), respectively) for each edge e ? G, then I(G) is decreasing (or increasing, respectively) with addition of edges. In this paper, we determine the extremal values of some monotonic topological indices which decrease or increase with addition of edges, and characterize the corresponding extremal graphs among bipartite graphs with a given connectivity.

Let G be a simple connected graph with n vertices and m edges, sequence of vertex degrees Δ = d1 ≥ d2 ≥...dn = δ > 0 and diagonal matrix D = diag(d1,d2, ..., dn) of its vertex degrees. Denote by Kf (G) = nΣn-1, i=1 1/μi , where μi are the Laplacian eigenvalues of graph G, the Kirchhoff index of G, and by NK = Πn,i=1 di the Narumi-Katayama index. In this paper we prove some inequalities that exhibit relationship between the Kirchhoff and Narumi-Katayama indices.

Let R be a commutative ring with identity and H be a nonempty proper multiplicative prime subset of R. The generalized total graph of R is the (undirected) simple graph ?GTH(R) with all elements of R as the vertex set and two distinct vertices x and y are adjacent if and only if x + y ? H. The complement of the generalized total graph ?GTH(R) of R is the (undirected) simple graphwith vertex set R and two distinct vertices x and y are adjacent if and only if x + y < H. In this paper, we investigate certain domination properties of ?GTH(R). In particular, we obtain the domination number, independence number and a characterization for -sets in ?GTP(Zn) where P is a prime ideal of Zn. Further, we discuss properties like Eulerian, Hamiltonian, planarity, and toroidality of GTP(Zn).