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Up to now there is no homotopy for Marcus-Wyse (for short $M$-) topological spaces. In relation to the development of a homotopy for the category of Marcus-Wyse (for short $M$-) topological spaces on ${\bf Z}^2$, we need to generalize the $M$-topology on ${\bf Z}^2$ to higher dimensional spaces $X \subset {\bf Z}^n$, $n \geq 3$ \cite{HL1}. Hence the present paper establishes a new topology on ${\bf Z}^n, n \in {\bf N}$, where ${\bf N}$ is the set of natural numbers. It is called the {\it generalized Marcus-Wyse} (for short $H$-) topology and is denoted by $({\bf Z}^n, \gamma^n)$. Besides, we prove that $({\bf Z}^3, \gamma^3)$ induces only $6$- or $18$-adjacency relations. Namely, $({\bf Z}^3, \gamma^3)$ does not support a $26$-adjacency, which is quite different from the Khalimsky topology for $3$D digital spaces. After developing an $H$-adjacency induced by the connectedness of $({\bf Z}^n, \gamma^n)$, the present paper establishes topological graphs based on the $H$-topology, which is called an $HA$-space in the paper, so that we can establish a category of $HA$-spaces. By using the $H$-adjacency, we propose an $H$-topological graph homomorphism (for short $HA$-map) and an $HA$-isomorphism. Besides, we prove that an $HA$-map ({\it resp.} an $HA$-isomorphism) is broader than an $H$-continuous map ({\it resp.} an $H$-homeomorphism) and is an $H$-connectedness preserving map. Finally, after investigating some properties of an $HA$-isomorphism, we propose both an $HA$-retract and an extension problem of an $HA$-map for studying $HA$-spaces.

The goal of the present paper is to study an extension problem of a connected preserving (for short, CP-) map between Khalimsky (K-for brevity, if there is no ambiguity) spaces. As a generalization of a K-continuous map, for K-topological spaces the recent paper [13] develops a function sending connected sets to connected ones (for brevity, an A-map: see Definition 3.1 in the present paper). Since this map plays an important role in applied topology including digital topology, digital geometry and mathematical morphology, the present paper studies an extension problem of a CP-map in terms of both an A-retract and an A-isomorphism (see Example 5.2). Since K-topological spaces have been often used for studying digital images, this extension problem can contribute to a certain areas of computer science and mathematical morphology.

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.

Classes of circulant graphs play an important role in modeling interconnection networks in parallel and distributed computing. They also find applications in modeling quantum spin networks supporting the perfect state transfer. It has been noticed that unitary Cayley graphs as a class of circulant graphs possess many good properties such as small diameter, mirror symmetry, recursive structure, regularity, etc. and therefore can serve as a model for efficient interconnection networks. In this paper we go a step further and analyze some other characteristics of unitary Cayley graphs important for the modeling of a good interconnection network. We show that all unitary Cayley graphs are hamiltonian. More precisely, every unitary Cayley graph is hamiltonian-laceable (up to one exception for X6) if it is bipartite, and hamiltonianconnected if it is not. We prove this by presenting an explicit construction of hamiltonian paths on Xnm using the hamiltonian paths on Xn and Xm for gcd(n,m) = 1. Moreover, we also prove that every unitary Cayley graph is bipancyclic and every nonbipartite unitary Cayley graph is pancyclic.

In the present paper, firstly, we review the notion of the SO-complete metric spaces. This notion let us to consider some fixed point theorems for single-valued mappings in incomplete metric spaces. Secondly, as motivated by the recent work of H. Baghani et al.(A fixed point theorem for a new class of set-valued mappings in R-complete (not necessarily complete) metric spaces, Filomat, 31 (2017), 3875-3884), we obtain the results of Ansari et al. [J. Fixed Point Theory Appl. (2017), 1145-1163] with very much weaker conditions. Also, we provide some examples show that our main theorem is a generalization of previous results. Finally, we give an application to the boundary value system for our results.

Let D(G) be the distance matrix and let ?1(D(G)) ? ... ? ?n(D(G)) be the corresponding eigenvalues of a connected graph G. Let m?(D) denote the multiplicity of the eigenvalue ? of the distance matrix D of G. In this paper, we characterize the graphs with m-2(D(G))= n-i, where i = 1,2,3,4. Furthermore, we show that both S+n and Sa,b (a+b=n-2) are determined by their D-spectrum.

In this paper, a new type of graph contractive multi-valued mappings in a metric space with a directed graph is introduced and studied. A common fixed point theorem of those two multi-valued mappings is established under some appropriate conditions. Moreover, some examples illustrating our main result are also given. The obtained result extends and generalizes several fixed point results of multivalued mappings in the literature. We apply our main result to obtain common fixed point results for two multi-valued mappings in ?-chainable complete metric spaces and two cyclic contraction multi-valued mappings.

The Randic index R(G) of a graph G is the sum of the weights (dudv)-1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randic indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex trees with the third for n ? 7, the fourth for n ? 10, the fifth and the sixth for n ? 11 maximum Randic indices. Recently, Li et al. [The Randic indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex trees with the seventh, the eighth, the ninth and the tenth for n ? 11 maximum Randic indices. In this paper, we correct the ordering for the Randic indices of trees obtained by Li et al., and characterize the trees with from the seventh to the sixteenth maximum Randic indices. The obtained extremal trees are molecular and thereby the obtained ordering also holds for molecular trees.

[EN] In this paper we introduce and study the concept of strong convergence in fuzzy metric spaces (X, M,*) in the sense of George and Veeramani. This concept is related with the condition Lambda M-t > 0(x, y, t) > 0, which frequently is required or missing in this context. Among other results we characterize the class of s-fuzzy metrics by the strong convergence defined here and we solve partially the question of finding explicitly a compatible metric with a given fuzzy metric., Valentn Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P (MINECO/FEDER, UE).

The purpose of this paper is to present some fixed point results in b-metric spaces using a contractive condition of Hardy-Rogers type with respect to the functional H. The data dependence of the fixed point set, the well-posedness of the fixed point problem, as well as, the Ulam-Hyres stability are also studied.

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect dominating set in $G$ is denoted by $\gamma_ {\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $ n \ge \gamma_ {\stackrel{}{11}}(G) \ge \gamma_ {\stackrel{}{12}}(G)\ge \ldots \ge \gamma_ {\stackrel{}{1\Delta}}(G)=\gamma(G)$ in order to indicate how far is $G$ from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, $\gamma_ {\stackrel{}{12}}(G)=\gamma (G)$. Among them, one can find cographs, claw-free graphs and graphs with extremal values of $\Delta(G)$.

The Hosoya index Z(G) of a graph G is defined as the total number of edge independent sets of G. In this paper, we extend the research of [J. Ou, On extremal unicyclic molecular graphs with maximal Hosoya index, Discrete Appl. Math. 157 (2009) 391-397.] and [Y. Ye, X. Pan, H. Liu, Ordering unicyclic graphs with respect to Hosoya indices and Merrifield-Simmons indices, MATCH Commun. Math. Comput. Chem. 59 (2008) 191-202.] and order the largest n - 1 unicyclic graphs with respect to the Hosoya index.

The resistance distance was introduced by Klein and Randic as a generalization of the classical distance. The Kirchhoff index Kf (G) of a graph G is the sum of resistance distances between all unordered pairs of vertices. In this paper we determine the extremal graphs with minimal Kirchhoff index among all n-vertex graphs with k cut vertices where 1 ? k < n/2.

In this paper, we recall the notion of intuitionistic fuzzy 2- normed space introduced by Mursaleen and Lohani \cite { muri } and using the direct method, we investigate the Hyers Ulam - Rassias stability of the following quadratic functional equation \begin {equation*} \ label{main} f(ax+by)+f(ax-by)-\frac{a}{2}f(x+y)=\frac{a}{2}f(x-y)-(2a^{2}-a)f(x)-(2b^{2}-a)f(y) \end {equation*} in this space.

In this paper we study fuzzy hyperideals of a fuzzy hyperring, we define and analyze two particular kinds of fuzzy hyperideals, which extend similar notions of ring context, namely prime fuzzy hyperideals and maximal fuzzy hyperideals. Moreover, we study the hyperideal transfer through a fuzzy hyperring homomorphism, particularly for prime fuzzy hyperideals and for maximal fuzzy hyperideals.

In [1], Abdo and Dimitov defined the total irregularity of a graph G=(V,E) as irrt(G)=1/2 ?u,v?V|dG(u)-dG(v)|, where dG(u) denotes the vertex degree of a vertex u ? V. In this paper, we investigate the minimal total irregularity of the connected graphs, determine the minimal, the second minimal, the third minimal total irregularity of trees, unicyclic graphs, bicyclic graphs on n vertices, and propose an open problem for further research.

Let $A,X,Y$ be bounded linear operators. In this paper, we present the explicit expression for the Moore--Penrose inverse of $A-XY$. In virtue of the expression of $(A+X)^\dag$, we get the upper bounds of $\|(A+X)^\dag\|$ and $\|(A+X)^\dag-A^\dag\|$.

The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contains uncertainties. In order to provide these soft algebraic structures, we introduce the concepts of SI-h-bi-ideals and SI-h-quasi-ideals of hemirings. The relationships between these kinds of soft intersection h-ideals are established. Finally, some characterizations of h-hemiregular, h-intra-hemiregular and h-quasi-hemiregular hemirings are investigated by these kinds of soft intersection h-ideals.

The aim of the presented paper is to study the fixed point theorems in complete and compact fuzzy metric spaces as improvement of some recent results (M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30 (1984) 1-9.For this purpose, the condition of the maximum type defined by altering distance is used. The research is illustrated by three examples.