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In this paper, we introduce a new concept of super edge-magic sequence (SEMS) of a super edge-magic graph (SEMG) with p vertices and q edges. The super edge-magic sequence of natural numbers is denoted by , 1 ≤ i ≤ q. This sequence need not to be monotonic. In this track, we also drive some families of super edge-magic graphs from fabrication of new super edge-magic sequences by considering additional parameter. We complete this paper by discussing the special case like monotonic sequences related to the super edge-magic sequence.

In 1975, Rhoades (9) generalized the classes of, operators of lp type and operators of Cesaro type by introducing an arbitrary infinite matrix A = (ank) using approximation numbers of a bounded linear operator. In the same paper Rhoades has proved that for each fixed matrix A satisfying the condition |an,2k−1| + |an,2k| ≤ M|ank| on the matrix A = (ank) for each n,k = 1,2,··· and each p,0 < p ≤ ∞ the set of A−p type operators is a linear space and raised an open question whether this condition is necessary to be a linear space? In this paper we have answered the question in negation. Further, we have introduced and studied the class A (s) −p of s-type cesp operators using s-number sequence and Cesaro sequence spaces. We have also shown that the class A (s) − p forms a quasi-Banach operator ideal. Moreover, the inclusion

Let G = (V,E) be a simple graph on the vertex set V . In a graph G, A set S ⊆ V is a dominating set of G if every vertex in V S is adjacent to some vertex in S. The domination number of a graph Gγ(G)] is the minimum size of a dominating set of vertices in G. The bondage number of a graph G[Bdγ(G)] is the cardinality of a smallest set of edges whose removal results in a graph with domination number larger than that of G. In this paper we establish domination number and the bondage number for some families of graphs. AMS Subject Classification: 05C69

In this paper, we investigate the properties of fuzzy soft sets and fuzzy soft maps in stsc-quantales. We define a L-fuzzy (K,E)-soft topology and a L-fuzzy (K,E)-soft closure spaces as a Hohle's sense. We study the relations between L-fuzzy (K,E)-soft topologies and L-fuzzy (K,E)-soft closure spaces. We give their examples.

In this paper three types of strong vertices are introduced.A necessary and sufficient conditions for a vertex to be strong have been given.Some of the properties of �- strong,�-strong and -strong vertices are discussed.Also the concept of vertex order colouring is introduced.An algorithm for vertex order colouring is also given at the end of the chapter.

In this paper, some theorems of edge regular fuzzy graphs are discussed with their complements and µ-complements. A necessary and sufficient condition under which they are equivalent is provided. Finally, adjacency sequence of edges in a fuzzy graph is defined. Using the sequences, characterization for a fuzzy graph with at most four vertices to be edge regular is obtained.

A clique (convex) dominating set S of G is a 1-movable clique dominating set (resp. 1-movable convex dominating set) of G if for every v ∈ S, either S \ {v} is a clique (resp. convex) dominating set or there exists a vertex u ∈ (V (G) \ S) ∩ NG(v) such that (S \ {v}) ∪ {u} is a clique (resp. convex) dominating set of G. The minimum cardinality of a 1-movable clique (resp. 1-movable convex) dominating set of G, denoted by 1 mcl(G) (resp. 1 mcon(G)), is called the 1-movable clique domination number (resp. 1-movable convex domination number) of G. A 1-movable clique dominating set in G with cardinality 1 mcl(G) is called a 1-set of G. This paper aims to characterize the 1-movable clique dominating sets of some graphs including those resulting from the join and composition of two graphs. The corresponding 1-movable clique domination number of the resulting graph is then determined. Further, it is shown that the concepts of 1-movable clique domination and 1-movable convex domination are equivalent.

Let C be a nonempty closed convex subset of a Hilbert space H, A : C → C be a nonexpansive mapping, B : C → H be a τ -inverse strongly monotone mapping and M be a maximal monotone operator on H such that the domain of M is included in C. In this paper, we prove the iterative sequence with errors converges weakly to a common element of F (A) and (B +M)0 under the suitable conditions. AMS Subject Classification: 47H05, 47H09, 46C05

In this paper, we define chromatic number of totally regular fuzzy graph and we introduce the concept of fuzzy total chromatic number of a totally regular fuzzy graph. Here we consider the fuzzy graphs with fuzzy sets of vertices and fuzzy sets of edges.

The concept of the Fibonacci number of an undirected graph G=(V,E) refers to the number of independent vertex subsets U of V such that no two vertices from U are adjacent in G. In this paper the Fibonacci numbers of molecular graphs corresponding to one type of phenylenes are calculated using the decomposition formula. Investigation of the Fibonacci numbers of certain classes of graphs leads to a difference equation or systems of difference equations. The explicit formula for the Fibonacci numbers of linear phenylenes is found as a function of the number n of hexagons in the phenylene. Pojem Fibonacciova čísla neorientovaného grafu G=(V,E) odpovídá počtu uzlově-nezávislých podmnožin U z V takových, že žádné dva uzly z U nejsou sousední v G. V tomto článku jsou použitím jisté dekompoziční formule určena Fibonacciova čísla molekulárních grafů odpovídajících jednomu typu fenylenů. Zkoumání Fibonacciových čísel jisté třídy grafů vede na diferenční rovnici nebo na soustavu diferenčních rovnic. Explicitní formule pro Fibonacciova čísla lineárních fenylenů je nalezena jako funkce počtu n hexagonů v tomto typu fenylenů.

The transitivity, primitivity, rank and subdegrees, as well as pairing of the suborbits associated with the action of the alternating group An, on unordered r−element subsets of a set X = {1, 2, · · · , n} of n letters, have not received any attention. In this paper, we prove that this action is transitive. We also show that the action is imprimitive if and only if n = 2r. In addition, we establish that the rank associated with the action is a constant r + 1 if and only if n ≥ 2r, except for r = 2 in which case the rank is 4 if n = 4, but is 3 for all n ≥ 5. Further, we calculate the subdegrees associated with the action and arrange them according to their increasing magnitudes. Finally, we show that all the suborbits of the action, with the exception of some non-trivial suborbits corresponding to the actions of A3 and A4 on the set of unordered pairs, are self-paired.

Let G be a finite non-abelian group and let Ω be a set of elements of G. Let A be the set of commuting elements in Ω, i.e A = {v ∈ Ω : vg = gv, g ∈ G}. In this paper, we extend the work on conjugate graph by defining a new graph called the orbit graph, denoted as ΓGΩ. The vertices of ΓGΩ are non central elements in Ω but not in A in which two vertices of ΓGΩ are adjacent if they are conjugate. Some graph properties are provided. Besides, the orbit graph of dihedral groups and quaternion groups is determined.

In this paper we have obtained cordial labeling for cycle of complete bipartite graphs and cycle of wheels. AMS Subject Classification: 05C78

A hyperoperation ◦ on a nonempty set H is a function from H×H into P ∗(H) where P ∗(H) is the set of all nonempty subset of H and (H, ◦) is call a hypergroupoid. A hypergroupoid (H, ◦) is called a semihypergroup if the hyperoperation ◦ is associative. Thus, semihypergroups generalize semigroups. Moreover, if S is a semigroup; we can define a hyperoperation ◦ on S in order to make (S, ◦) a semihypergroup. In 2013, R.I. Sararnrakskul defined a hyperoperation ◦ on the partial transformation semigroup P (X) to make a semihypergroup. In this paper, we define a regular equivalence relation ρ on (P (X), ◦) so that P (X)/ρ is a semihypergroup and then we studies some subsemihypergroup of P (X)/ρ. AMS Subject Classification: 20M20, 20N20 Received: November 10, 2014 c © 2015 Academic Publications, Ltd. url: www.acadpubl.eu §Correspondence author 22 R.I. Sararnrakskul, S. Pianskool

An image of a plane graph, G = (V,E) of order n and size m, is said to be a vertex-edge-magic plane graph if there is a bijection f : V ∪ E → {1,2,..,n+m} such that for all s−side faces of G, except the infinite face, the sum of the labels of its vertices and edges is a constant k(s). Such a bijection will be called a vertex-edge-magic plane labeling of G. In case that all the finite sides of a graph G having the same size we will be interested in determining the minimum and the maximum number, k, such that there exists a vertex-edge- magic labeling of G, in which k is the sum of the vertex and edge labeling of each face. In this paper we find such a minimum and maximum numbers for a wheel with even order.

In this paper, we define the dominant set, zero divisor of the rough semiring (T,�,∇). Also We prove that RS(X) is not a zero divisor for a dominant set X ⊆ U where U is the finite universal set on the set of all rough sets for the given information system together with the operations prabaand praba ∇. We illustrate these concepts through examples.

In this paper we introduce the Max-Min intuitionistic fuzzy matrix M (G) of an intuitionistic fuzzy graph. And the extreme energy of M (G) is defined. And also we give the explicit expression for the coefficients of the characteristic polynomial of M (G) . These concepts are illustrated with real time example.

This research note constructs S-H fuzzy collections and S-H fuzzy partitions of a finite set. The main objective of the paper lies in defining two different operations on a class of S-H fuzzy partitions of a set and, in turn, proving that these give rise to a monoid. AMS Subject Classification: 03E72, 46S50, 06F05

Let � ' (G) and � ' (G) be the matching and edge covering number , respectively. The strong product G1⊠G2 of graph of G1 and G2 has vertex set V (G1⊠G2) = V (G1)×V (G2) and edge set E(G1⊠G2) = {(u1,v1)(u2,v2)|(u1u2 ∈ E(G1) and v1v2 ∈ E(G2)) or (u1 = u2 and v1v2 ∈ E(G2)) or (u1u2 ∈ E(G1) and v1 = v2)}. In this paper, we determined generalization of matching number and edge covering number on strong product of complete bipartite graphs and any simple graph.

Let G be a group. Let R be a commutative G-graded ring, M be a graded R-module and N be a graded R-submodule of M. In this paper, we study some cases when R is strongly graded ring and the component Ne of N is strictly non-prime, pseudo prime or pseudo irreducible Re-submodule. For example, we prove that if R is strongly graded, the components of M are multiplication and Ne is pseudo irreducible, then Ng is pseudo prime for all g ∈ G.

In an earlier paper the author has shown that a graph is recon- structible, except possibly if it has a single-block trunk. Here this result is shown for colored graphs. Some other results on colored graphs are given. In particular it is shown that reconstructibility is false for edge colored graphs.

Let Gn,d be a random d-regular graph with n vertices. Given a fixed graph H. W denotes the number of induced copies of H in Gn,d. In this paper, we use Stein-Chen method and Local approach to show that W can approximate by the Poisson distribution and give the bound of this approximation.

Let G be a fuzzy graph. Let u and v be two vertices of G. A subset D of V is called a fuzzy equitable dominating set if every v ∈ V −D there exist a vertex u ∈ D such thatuv ∈ E(G) and |deg(u) − deg(v)| ≤ 1 where deg(u) denotes the degree of vertex u and deg(v) denotes the degree of vertex v and µ(uv) ≤ �(u)∧�(v). The minimum cardinality of a fuzzy equitable dominating set is denoted by ef . In this paper we introduce the concept of fuzzy equitable dominating set, minimal fuzzy equitable dominating set, strong (weak) fuzzy equitable dominating set , fuzzy equitable independent set and obtain some interesting results for this new parameter in equitable fuzzy graphs.

Let Wn,r be the number of isolated complete graphs of order r in a random intersection graph G(n,m,p). In this paper, we demonstrate that Wn,r can be approximated by Poisson distribution and give the bound of this approximation by using the Stein-Chen method.

In this paper some equivalence relation in the class of bounded sequences of positive numbers is discussed. This relation enables to understand better the permutations preserving divergence of the same rate, introduced by Johnston. It also gives the possibility to prove, in a new way, the combinatoric characterization of these permutations.

The notion of matching in a fuzzy graph could be defined using the concept of effective edges (8) or by fractional matching (4). In this paper, we derive a necessary condition for a fuzzy graph on a cycle or a complete graph or a stargraph to have a perfect fuzzy matching. Also we discuss perfect fuzzy matching on strong regular fuzzy graphs.

Let Wn,k be the number of isolated cycles of order k in a random intersection graph G(n,m,p). In this paper, we demonstrate that Wn,k can be approximated by Poisson distribution and give the bound of this approximation by using the Stein-Chen method.

A subset of vertices of a graph G is called a feedback vertex set of G if its removal results in an acyclic subgraph. The minimum cardinality of a feedback vertex set is called the feedback number. In this paper, we investigate the feedback number of flower snark and related graphs Hn. Let f(Hn) denote the feedback number of Hn, we prove that

In this paper, we show that B 2 (n,n) is harmonious (4) , P n i is harmonious (3), Pn × Cm is vertex graceful for n ≥ 2, and m ≥ 5,m is odd, B 2 (n,n) is vertex graceful (4) , P n

A graph G has an equitable k-defective coloring in m colors if its vertices can be colored with m colors such that the maximum degree of any subgraph induced by vertices assigned to the same color is at most k and the numbers of vertices in any two sets composed of the vertices that are assigned to the same color differ by at most one. The equitable k-defective chromatic number of a graph G, denoted by χED,k(G), is the smallest positive integer m for which G has an equitable k-defective coloring in m colors. In this paper, we present the equitable k-defective chromatic numbers of complete bipartite graphs for k = 1 and k = 2.

In this paper we introduce fuzzy det-norm ordering with fuzzy matrices using the structure of Mn(F), the set of (n×n) fuzzy det-norm ordering with fuzzy matrices is introduced. From this row and column, determinant of the fuzzy norm has been obtained by imposing an equivalence relation on Mn(F). We know that the comparability relation on fuzzy matrices is a partial ordering. We prove that det-norm ordering is a partitions ordering on the set of all idempotent matrices in Mn(F). We begin with the det-norm ordering on fuzzy matrices as analogue of the ordering on real matrices. Several properties of these orderings are derived. Discuss their relationship between these ordering with det-norm ordering and Also we introduce the concept of Fuzzy norm and partitions of Mn(F), Properties of fuzzy det-norm ordering.

Let W be the number of copies of a fixed graph H in a random dregular graph Gn,d. In this paper, we give the bound on Poisson approximation of W by using the Stein-Chen method.

Square Cycle, C 2 n is a graph that has n vertices and two vertices u and v are adjacent if and only if distance between u and v not greater than 2. In this paper, we show that the determinant of adjacency matrix of square cycle C 2 n are as follows

The cycle-structure of hypergraph plays an important role in the hypergraph theory and the database theory. It has been well studied during last several years. Especially in recent years, many researchers continue to study. Wang introduced the cycle-axiom of hypergraph. In this paper, we make a note of cycle-axiom of hypergraph to explain that the cycle-axiom of hypergraph is reasonable, and displays an important role in the hypergraph theory.

An image of a plane graph, G = (V,E) of order n and size m, is said to be an edge-magic plane graph if there is a bijection f : E → {1,2,..,m} such that for all s−side faces of G, except the infinite face, the sum of the labels of its edges is a constant k(s). Such a bijection will be called an edge-magic plane labeling of G. In case that all the finite sides of a graph G having the same size we will be interested in determining the minimum and the maximum number, k, such that there exists an edge-magic plane labeling of G, in which k is the sum of the edge labeling of each face. In this paper we find such a minimum and maximum numbers for a wheel with even order. Furthermore we conjecture that the same formula is valid for the odd case.

In this paper, we introduce a new class of hanging theta graphs and prove that those graphs admit graceful labeling.

In this paper, we introduce an equivalence relation B on an ordered semigroup S. Using the relation B defined, we characterize 0-minimal (m,n)ideals of S. Several sufficient conditions are presented for 0-minimal left ideals and 0-minimal right ideals of S to be 0-minimal (m,n)-ideals of S. The results obtained generalize the results on semigroups without order. AMS Subject Classification: 06F05

In this paper, the consecutive even number finding graph has been discussed. The solution of Goldbach conjecture has been established from this graph.

A set S of vertices of a graph G = (V,E) is called a vertex cover, if each edge in E has at least one end point in S and the minimum cardinality taken over all vertex covering sets of G is called the covering number of G denoted by �(G). In this paper, we study some results on vertex covering, edge covering, strong and weak covering, and inverse covering numbers denoted by �(G), � ' (G), s�(G), w�(G), � −1 (G) respectively for generalized Jahangir graphs Js,m. We have also characterized the graphs which are invertible.

A set D of a vertices in a graph G = (V,E) is said to be a total dominating set of G if every vertex in V is adjacent to some vertex in D. The total domination number t(G) is the minimum cardinality of a total dominating set. If t(G) 6 |V (G)|, the minimum cardinality of a set E0 ⊆ E(G), such that G − E0 contains no isolated vertices and t(G − E0) > t(G), is called the total bondage number of G. This paper determines the exact values of total bondage number of Wheel graph, Helm graph, Windmill graph, Circular necklace and Friendship graph.

The packing chromatic number ��(G) of a graph G is the smallest integer k for which there exists a mapping � : V (G) −→ {1,2,...,k} such that any two vertices of color i are at distance at least i + 1. It is a frequency assignment problem used in wireless networks, which is also called broadcasting coloring. It is proved that packing coloring is NP-complete for general graphs and even for trees. In this paper, we study the packing chromatic number of comb graph, circular ladder, windmill, H-graph and uniform theta graph.

In this paper, we investigate the properties of pairs of interval negations and interval implications on a bounded lattice. In particular, we study the relations between pairs of negations and implications and pairs of interval negations and interval implications.

In this paper, we prove that the D2(Pn) is an even sequential harmonious and graceful graph, D2(K1, n) an even sequential harmonious, graceful graph, odd graceful and felicitous graph, spl(Pn) is an even sequential harmonious and odd graceful graph, spl(K1, n) is an even sequential harmonious and felicitous graph. AMS Subject Classification: even sequential harmonious, graceful, odd graceful, felicitous shadow graph, split graph

In this paper, we use Stein-Chen and Coupling method to give non- uniform bounds in Poisson approximation of the number of isolated vertices in a random intersection graph.

In this paper first, we give a brief introduction about integer distance graphs. An integer distance graph is a graph G(Z,D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if |u − v| ∈ D where D is a subset of the positive integers. If D is a subset of P then we call G(Z,D) a prime distance graph. Second, we obtain a partial solution to a general open problem of characterizing a class of prime distance graphs. Third, we compute the vertex arboricity of certain prime distance graphs. Fourth, we give a brief review regarding circulant graphs and highlight its importance in the computation of chromatic number of distance graphs with appropriate references. Fifth, we introduce the notion of pseudochromatic coloring and obtain certain results concerning circulant graphs and distance graphs.

Let O[n] and C[n] be the semigroup of all order-preserving and all order-preserving and regressive transformations on the set [n] = {1, . . . , n}, respectively. In this paper, we give a pattern on [n]. Combinatorial properties on subsemigroups of O[n] with restricted range to the pattern are investigated. AMS Subject Classification: 20M20, 11B39

Let (G) and ' (G) be the domination number and edge dom- ination number, respectively. The lexicographical Product G1 • G2 of graph of G1 and G2 has vertex set V (G1 • G2) = V (G1) × V (G2) and edge set E(G1 •G2) = {(u1v1)(u2v2)| (u1u2 ∈ E(G1))∪(u1 = u2 and v1v2 ∈ E(G2))}. In this paper, we determined generalization of domination and edge domination number on lexicographical product of complete graphs and any simple graph.

In this paper we extend the concept of generalized Hamming weights for poset-weight codes and show that any linear code C satisfies the chain condition if support of C is a totally ordered subposet.

Let � ' (G) and � ' (G) be the matching number and edge covering number of G, respectively. The tensor product G1 ⊗ G2 of graph of G1 and G2 has vertex set V (G1 ⊗ G2) = V (G1) × V (G2) and edge set E(G1 ⊗ G2) = {(u1v1)(u2v2) | u1u2 ∈ E(G1) and v1v2 ∈ E(G2)}. In this paper, we determined generalization of matching number and edge covering on tensor product of fan graph and any simple graph.

Let � ' (G) and � ' (G) be the matching and edge covering num- ber , respectively. The lexicographical product G1 • G2 of graph of G1 and G2 has vertex set V (G1 • G2) = V (G1) × V (G2) and edge set E(G1 • G2) = {(u1v1)(u2v2)|(u1u2 ∈ E(G1)) ∪ (u1 = u2 and v1v2 ∈ E(G2))}. In this paper, we determined generalization of matching and edge covering number on lexico- graphical product of complete graphs and any simple graph.