Search

Abstract: Upward-closed sets of integer vectors enjoy the merit of having a finite number of minimal elements, which is behind the decidability of a number of Petri net related problems. In general, however, such a finite set of minimal elements may not be effectively computable. In this paper, we develop a unified strategy for computing the sizes of the minimal elements of certain upward-closed sets associated with Petri nets. Our approach can be regarded as a refinement of a previous work by Valk and Jantzen (in which a necessary and sufficient condition for effective computability of the set was given), in the sense that complexity bounds now become available provided that a bound can be placed on the size of a witness for a key query. The sizes of several upward-closed sets that arise in the theory of Petri nets as well as in backward-reachability analysis in automated verification are derived in this paper, improving upon previous decidability results shown in the literature. [Copyright &y& Elsevier]

Abstract: An incidence graph of a given graph G, denoted by I(G), has its vertex set such that the pair of vertices is an edge of I(G) if and only if there exists at least one case of , , or . Study of Incidence graphs was made by Zhagn Zhong-fu et al. in [Zhagn Zhong-fu, Yao Bing, Li Jing-wen, Linu Lin-zhong, Wang Jian-fang, Xu Bao-gen, On Incidence Graphs, ARS combinatoria 87 (2008), 213-223]. The origin of Incidence graphs can be traced to a paper titled Incidence and Strong edge colorings of graphs by R.A. Brualdi, et al. [Richard A. Brualdi and Jennifer J. Quinn Massey, Incidence and strong edge colorings of graphs, Discrete Mathematics 122 (1993) 51-58]. In this paper, we make a study of domination and related parameters in Incidence graphs. [Copyright &y& Elsevier]

Abstract: N.S. Bhave and B.Y. Bam [S. S. Kamath and Saroja R. Hebbar, Strong and weak Domination, Full sets and Domintion Balance in Semigraphs, Preprint], introduced the concept of middle end degree ((m, e)− degree) of a vertex in a semigraph. This paper is devoted to a particular type of a-domination in which the (m, e)− degree plays a part. A vertex u (m, e)-strong dominates another vertex v if u and v are adjacent and in the lexico graphic order. (m, e)- dominating sets and (m, e)- domination number are defined and studied in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we study the following second-order delay differential equation with mixed nonlinearitieswhere . Oscillation criteria for the equation are established which generalize the results by Džurina and Stavroulakis [J.D. Džurina, I.P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 140 (2003) 44–453] and by Sun and Meng [Y.G. Sun, F.W. Meng, Note on the paper of Džurina and Stavroulakis, Appl. Math. Comput. 174 (2006) 1634–1641]. [Copyright &y& Elsevier]

Abstract: There are three types of distributional chaos, namely DC1, DC2 and DC3. In this paper we present two constant-length substitution systems, one is DC2 but not DC1, and the other is DC3 but not DC2. (In this paper, chaos means existence of an uncountable scrambled set of the corresponding type while the existing examples deal with single pairs of points only.) [Copyright &y& Elsevier]

Abstract: In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers and , where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts. The numbers and , generalize the generalizations to coincidences, of Zhao''s Nielsen number on the complement , respectively Schirmer''s relative Nielsen number . Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps and which are homotopic as maps of M -ads to f, respectively k firstly on the complement of the union of the subspaces in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer''s triad Nielsen number . Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the study of necessary and sufficient optimality conditions for convex–concave generalized fractional disjunctive programming problems for which the decision set is the union of a family of convex sets. The Lagrangian function for such problems is defined and the Kuhn–Tucker Saddle and Stationary points are characterized. In addition, some important theorems related to the Kuhn–Tucker problem for saddle and stationary points are established. Moreover, a general dual problem is formulated and weak, strong and converse duality theorems are proved. Throughout the presented paper illustrative examples are given to clarify and implement the developed theory. [Copyright &y& Elsevier]

Abstract: Intersection types are well known to type theorists mainly for two reasons. Firstly, they type all and only the strongly normalizable lambda terms. Secondly, the intersection type operator is a meta-level operator, that is, there is no direct logical counterpart in the Curry–Howard isomorphism sense. In particular, its meta-level nature implies that it does not correspond to the intuitionistic conjunction. The intersection type system is naturally a type inference system (system à la Curry), but the meta-level nature of the intersection operator does not allow to easily design an equivalent typed system (system à la Church). There are many proposals in the literature to design such systems, but none of them gives an entirely satisfactory answer to the problem. In this paper, we will review the main results in the literature both on the logical interpretation of intersection types and on proposed typed lambda calculi. The core of this paper is a new proposal for a true intersection typed lambda calculus, without any meta-level notion. Namely, any typable term (in the intersection type inference) has a corresponding typed term (which is the same as the untyped term by erasing the type decorations and the typed term constructors) with the same type, and vice versa. The main idea is to introduce a relevant parallel term constructor which corresponds to the intersection type constructor, in such a way that terms in parallel share the same resources, that is, the same context of free typed variables. Three rules allow us to generate all typed terms. The first two rules, Application and Lambda-abstraction, are performed on all the components of a parallel term in a synchronized way. Finally, via the third rule of Local Renaming, once a free typed variable is bounded by lambda-abstraction, each of the terms in parallel can do its local renaming, with type refinement, of that particular resource. [Copyright &y& Elsevier]

Abstract: In previous papers, we introduced a new reasoning method based on quantifying linguistic domains, established a new fuzzy control algorithm, called hedge-algebras-based controller (HAC), and applied it to solve some fuzzy control problems. The HAC does not require fuzzy sets to provide the semantics of the linguistic terms used in the fuzzy rule system rather the semantics is obtained through the semantically quantifying mappings (SQMs). It was shown that the new method is very effective, i.e. for these problems it is always able to efficiently control the process toward the stable state. In the algebraic approach, the design of an HAC leads to the determination of the parameters of SQMs, which are the fuzziness measure of primary terms and linguistic hedges occurring in the fuzzy model, and the weights of a weighted averaging operator. However, there exists another problem, namely, how one can determine the optimal parameters of the method. In the present paper, we improved the HAC''s design by adding an optimal step that aims to find the optimal parameters of HAC using a genetic algorithm (GA). To show the effectiveness of the proposed method, we apply it to solve again the inverted pendulum problem and the problem of holding an object on a “hill”. The results demonstrate the good performance of the designed HAC. [Copyright &y& Elsevier]

Abstract: This paper shows that the greatest and least monotone extensions of a null-additive (resp. weakly null-additive), monotone set function from a ring of subsets to the algebra generated by the ring are null-additive (resp. weakly null-additive). In addition, the paper characterizes all the (weakly) null-additive, monotone extensions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the study of necessary and sufficient optimality conditions for convex–concave fractional disjunctive programming problems for which the decision set is the union of a family of convex sets. The Lagrangian function for such problems is defined and the Kuhn–Tucker saddle and stationary points are characterized. In addition, some important theorems related to the Kuhn–Tucker problem for saddle and stationary points are established. Moreover, a general dual problem is formulated, and weak, strong and converse duality theorems are proved. Throughout the presented paper illustrative examples are given to clarify and implement the developed theory. [Copyright &y& Elsevier]

Abstract: The paper introduces a simple, yet powerful axiomatization of Zadeh''s notion of fuzzy set, based on formal fuzzy logic. The presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics. Its essence is elementary fuzzy set theory, cast as two-sorted first-order theory over fuzzy logic, which is generalized to simple type theory. We show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism. In this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed; however, we also demonstrate how to add structure to the universe of discourse within our framework. [Copyright &y& Elsevier]

Abstract: The paper gives a concrete presentation of the minimal structure on the monoid of an external subset classifier of a universe of usual sets with Zadeh–Goguen L-fuzzy subsets. The result was first investigated by the first author, with the aim of providing a new insight into the synchronic approach to fuzzy set theory. The extensive use of conceptual mathematics (i.e. category theory) in the original paper, restricts the audience of readers of this analysis to persons who are already acquainted with conceptual mathematics. The aim of this paper is to avoid as much as possible the use of functorial terminology in the presentation of the minimal mathematical structure on the set of truth values, for a synchronic approach to fuzzy sets. By doing so, we hope to widen the audience of scientists interested in the foundation of Zadeh–Goguen L-fuzzy sets and to provide a new way of looking at the synchronic approach to fuzziness. [Copyright &y& Elsevier]

Abstract: Let . For or , has been determined by Hanani. In this paper, we investigate the case of . It is easy to see that if , then . It is known that by Hanani and that by a previous paper of the author. We shall focus on the case of . It is proved that and . [Copyright &y& Elsevier]

A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs: (1)Pn, a path on n vertices;(2)K1,ns, the graph obtained from K1,n by subdividing an edge once;(3)K2,n&z.drule;e, the graph obtained from K2,n by deleting an edge;(4)K2,n+, the graph obtained from K2,n by adding an edge between the two degree-n vertices x1 and x2, and a pendent edge at each xi.Two applications of this result are also discussed in the paper. [Copyright &y& Elsevier]

Let X and Y be topological spaces. We say that X × Y satisfies the Uniquely Universal property (UU) iff there exists an open set U ⊆ X × Y such that for every open set W ⊆ Y there is a unique cross section of U with U ( x ) = { y ∈ Y : ( x , y ) ∈ U } = W . Arnold W. Miller in his paper [1] posed the following two questions: Problem 1 Does [ 0 , 1 ] × ω have UU? Problem 2 Does R × ω have UU? In this paper we present two constructions which give positive answers to both problems. [ABSTRACT FROM AUTHOR]

A half-space over a distance space is a generalization of a half-space in a vector space. An important advantage of a distance space over a metric space is that the triangle inequality need not be satisfied, which makes our results potentially very useful in practice. Given two points in a set, a half-space is defined by them, as the set of all points closer to the first point than to the second. In this paper we consider the problem of learning half-spaces in any finite distance space, that is, any finite set equipped with a distance function. We make use of a notion of ‘width’ of a half-space at a given point: this is defined as the difference between the distances of the point to the two points that define the half-space. We obtain probabilistic bounds on the generalization error when learning half-spaces from samples. These bounds depend on the empirical error (the fraction of sample points on which the half-space does not achieve a large width) and on the VC-dimension of the effective class of half-spaces that have a large sample width. Unlike some previous work on learning classification over metric spaces, the bound does not involve the covering number of the space, and can therefore be tighter. [ABSTRACT FROM AUTHOR]

In this paper we compute the critical group of the Kneser graph K G ( n , 2 ) . This is equivalent to computing the Smith normal form of a Laplacian matrix of this graph. [ABSTRACT FROM AUTHOR]

A space X is selectively absolutely star-Lindelöf [1,3] if for each open cover U of X and any sequence ( D n : n ∈ ω ) of dense subsets of X , there are finite sets F n ⊆ D n ( n ∈ ω ) such that S t ( ⋃ n ∈ ω F n , U ) = X . In this paper, we continue to investigate topological properties of selectively absolute star-Lindelöf spaces, and show the following statements: (1) There exists a Tychonoff selectively a-star-Lindelöf, pseudocompact space X having a regular closed G δ subset which is not star-Lindelöf (hence not selectively a-star-Lindelöf); (2) Assuming 2 ℵ 0 = 2 ℵ 1 , there exists a normal selectively a-star-Lindelöf space X having a regular closed G δ subset which is not star-Lindelöf (hence not selectively a-star-Lindelöf); (3) An open F σ -subset of a selectively a-star-Lindelöf space is selectively a-star-Lindelöf; (4) For any cardinal κ , there exists a Tychonoff selectively a-star-Lindelöf, pseudocompact space X such that e ( X ) ≥ κ . [ABSTRACT FROM AUTHOR]

The class of hyper-CR Einstein–Weyl structures on R 3 can be described in terms of the solutions to the dispersionless Hirota equation. In the present paper we show that simple geometric constructions on the associated twistor space lead to deformations of the Hirota equation that have been introduced recently by B. Kruglikov and A. Panasyuk. Our method produces also the hyper-CR equation and can be applied to other geometric structures related to different twistor constructions. [ABSTRACT FROM AUTHOR]

In this paper we prove the linear instability for the Boussinesq equations experiencing a diabatic forcing. To do so, we first prove the local well-posedness of the governing set of equations. An example of an unstable solution to the linearized set of equations is then constructed. Finally, we show that the unstable solution to the linearized problem is also unstable in the nonlinear sense. [ABSTRACT FROM AUTHOR]

A connected graph G with at least 2 m + 2 n + 2 vertices is said to satisfy the property E ( m , n ) if G contains a perfect matching and for any two sets of independent edges M and N with | M | = m and | N | = n with M ∩ N = ∅ , there is a perfect matching F in G such that M ⊂ F and N ∩ F = ∅ . In particular, if G is E ( m , 0 ) , we say that G is m -extendable. One of the authors has proved that every m -tough graph of even order at least 2 m + 2 is m -extendable (Plummer, 1988). Chen (1995) and Robertshaw and Woodall (2002) gave sufficient conditions on binding number for m -extendability. In this paper, we extend these results and give lower bounds on toughness and binding number which guarantee E ( m , n ) . [ABSTRACT FROM AUTHOR]

The Szeged index of a graph G is defined as S z ( G ) = ∑ e = u v ∈ E n u ( e ) n v ( e ) , where n u ( e ) and n v ( e ) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u . A cactus is a graph in which any two cycles have at most one common vertex. Let C ( n , k ) denote the class of all cacti with order n and k cycles, and C n t denote the class of all cacti with order n and t pendant vertices. In this paper, a lower bound of the Szeged index for cacti of order n with k cycles is determined, and all the graphs that achieve the lower bound are identified. As well, the unique graph in C n t with minimum Szeged index is characterized. [ABSTRACT FROM AUTHOR]

In this paper we enumerate and give bijections for the following four sets of vertices among rooted ordered trees of a fixed size: (i) first-children of degree k at level ℓ , (ii) non-first-children of degree k at level ℓ − 1 , (iii) leaves having k − 1 elder siblings at level ℓ , and (iv) non-leaves of outdegree k at level ℓ − 1 . Our results unite and generalize several previous works in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study four games, they are all restrictions of ( s , t ) -Wythoff’s game which was introduced by A.S. Fraenkel. The first one is a modular type restriction of ( s , t ) -Wythoff’s game, where a player is restricted to remove a multiple of K tokens in each move ( K is a fixed positive integer). The others we called rook type restrictions of ( s , t ) -Wythoff’s game, including Odd-Arbitrary-Nim ( s , t ) -Wythoff’s Game, Odd–Odd-Nim ( s , t ) -Wythoff’s Game and Odd–Even-Nim ( s , t ) -Wythoff’s Game. In these three games, the restrictions are only made on horizontal and vertical moves, but not on the extended diagonal moves. For any K , s , t ≥ 1 , the sets of P -positions of our games are given in both normal and misère play. [ABSTRACT FROM AUTHOR]

Entropy-based measures have been used to characterize the structure of complex networks. Various graph entropies based on different graph parameters have been proposed and studied. So, it has been intricate to decide which measure is the right one for solving a particular problem. In this paper, we introduce graph entropy measures based on independent sets and matchings of graphs. The values of entropies of some special graphs are calculated and we draw several conclusions regrading the usability of the measures. Based on our findings, we conclude that it should be demanding studying extremal values of the novel entropies. [ABSTRACT FROM AUTHOR]

It is obtained a selection theorem for paraconvex-valued mappings possessing two natural properties. Several known results follow easily from this theorem. Moreover, it implies the affirmative solution to a question stated in a paper of N.R. Loufouma Makala. [ABSTRACT FROM AUTHOR]

The paper contains new results on the impact of harvesting times and intensities on the stability properties of Seno population models. It is proved that sufficiently high harvest intensities are stabilizing for any harvesting time in the sense that they create a positive equilibrium that attracts all positive solutions. Moreover, in the special case that the nonlinearity in the Seno model is a Ricker function, we derive a global stability result independent of timing and valid for low to medium harvesting efforts. The proof is based on a characterization of those harvesting intensities which guarantee a negative Schwarzian derivative for all harvesting times. Finally, we rigorously show that timing can be stabilizing as well as destabilizing by itself. In particular, a recent conjecture formulated by Cid et al. (2014) [1] is shown to be false. [ABSTRACT FROM AUTHOR]

The (bi)simulation relation has recently been attracting growing interest in the study of nonlinear control systems, in the hope that through such a relation, the behaviors and properties of a nonlinear system can be inferred from those of another system which is easier to handle. In this paper, we consider the propagation of the property of nonlinear norm-observability through a simulation relation. Given two control systems that are related by a graph simulation relation, we derive conditions under which the norm-observability of the simulating system implies the norm-observability of the simulated system. The obtained results are given in terms of set-valued functions. Several examples are included to illustrate various applications of our results. [ABSTRACT FROM AUTHOR]

A Hamilton ℓ -cycle in a k -uniform hypergraph of n -vertex is an ordering of all vertices, combined with an ordered subset C of edges, such that any two consecutive edges share exactly ℓ vertices and each edge in C contains k consecutive vertices. A classic result of O. Ore in 1960 is that if the degree sum of any two independent vertices in an n -vertex graph is at least n , then the graph contains a Hamiltonian cycle. Naturally, we consider to generalize it to hypergraphs situation. In this paper, we prove the following theorems. (i) For any n ≥ 4 k 2 , there is an n -vertex k -uniform hypergraph, with degree sum of any two strongly independent sets of k − 1 vertices bigger than 2 n − 4 ( k − 1 ) , contains no Hamilton l -cycle, 1 ≤ ℓ ≤ k − 1 . (ii) If the degree sum of two weakly independent sets of k − 1 vertices in an n -vertex k -uniform hypergraph is ( 1 + o ( 1 ) ) n , then the hypergraph contains a Hamilton ( k − 1 ) -cycle, where two distinct sets of k − 1 vertices are weakly (strongly) independent if there exist no edge containing the union of them (intersecting both of them). [ABSTRACT FROM AUTHOR]

Bierbrauer (2012) developed the theory of q -linear cyclic codes over ( F q ) m and he obtained a parametric description of such codes by cyclotomic cosets. Recently, Cao et al. (2015) obtained the structure of cyclic additive codes over the Galois ring GR ( p ℓ , m ) , where m is a prime integer. Let R be a finite commutative ring and R n = R [ x ] ∕ 〈 x n − 1 〉 . In this paper, we generalize the theory of F q -linear codes over vector spaces to R -linear codes over free R -algebras (free as R -module). We call these codes, R -additive codes. We introduce a one-to-one correspondence between the classes of cyclic R -additive code and the classes of R n -linear code. Using the structure of R n -linear codes, we present the structure of cyclic R -additive codes, where R is a chain ring. Among other results, q -linear cyclic codes over ( F q ) m are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR ( p ℓ , m ) is given for an arbitrary integer m , not necessarily a prime number. [ABSTRACT FROM AUTHOR]

The class of Boolean functions, which can never occur as output of a faulty combinational circuit, is known as Impossible Class of Faulty Functions (ICFF). Despite years of research, the characterization of ICFF for a complex logic network is yet a significantly open problem. In a recent work (Das et al., 2014), a partial characterization of ICFF was done by completely enumerating a subclass of ICFFs, also called the root functions . For 2 -level AND-OR logic networks for up to 6 -variable the root functions were completely enumerated. This paper significantly extends this line of research by providing a more general framework for handling this class of functions. Both the upper and lower bounds are derived and the maximum cardinality of the weight of non-affine root functions is established. A constructive method for obtaining non-affine root functions of maximum possible weight is provided and it is shown that there are no other such functions than those obtained by our method. The direct correspondence of these functions to the independent dominating sets in the n -dimensional hypercube G n is used for establishing new results concerning the minimum weight of these functions. In particular, we give the exact value of the minimum cardinality of independent dominating set in G n for any n = 2 r − 1 , where r ≥ 2 . Since our approach is also constructive, we essentially efficiently solve this problem for any n = 2 r − 1 , where r ≥ 2 , and our results conform to the observation of Harary and Livingston (2010). Finally, the problem of classifying and enumerating root functions is addressed leading to some non-trivial lower bounds on their number, though the problem appears to be hard and further research in this direction is needed. [ABSTRACT FROM AUTHOR]

Let E be a set of points in F q d . Bennett et al. (2016) proved that if | E | ≫ q d − d − 1 k + 1 then E determines a positive proportion of all k -simplices. In this paper, we give an improvement of this result in the case when E is the Cartesian product of sets. Namely, we show that if E is the Cartesian product of sets and q k d k + 1 − 1 ∕ d = o ( | E | ) , the number of congruence classes of k -simplices determined by E is at least ( 1 − o ( 1 ) ) q k + 1 2 , and in some cases our result is sharp. [ABSTRACT FROM AUTHOR]

A space X is selectively absolutely star-Lindelöf if for any open cover U of X and any sequence ( D n : n ∈ ω ) of dense subsets of X , there are finite sets F n ⊂ D n ( n ∈ ω ) such that S t ( ⋃ n ∈ ω F n , U ) = X . This notion was introduced by S. Bhowmik [3] , and it lies between absolute countable compactness in Matveev [9] and absolute star-Lindelöfness in Bonanzinga [4] . In this paper, we distinguish absolute star-Lindelöfness from selective absolute star-Lindelöfness, and study the general properties of selectively absolutely star-Lindelöf spaces. [ABSTRACT FROM AUTHOR]

The purpose in this paper is to prove the existence of a canonical-pre-quasi-uniform extension ( X , U ) ˆ of a pre-quasi-uniform space ( X , U ) . Most of the concepts used in quasi-uniform spaces may be applied to pre-quasi-uniform spaces. These spaces were introduced by the first author and developed by the second in his doctoral thesis. We shall study the main properties of this class of spaces which contains the class of quasi-uniform spaces. [ABSTRACT FROM AUTHOR]

A dihedral SQS is an SQS admitting a point-regular dihedral automorphism group. Some constructions of dihedral SQSs are established in this paper. We also prove that for v ≡ 2 , 4 ( m o d 6 ) , there is a dihedral SQS ( v ) if there is a dihedral H ( p , 2 , 4 , 3 ) design or an S -cyclic SQS ( 2 p ) for each odd prime divisor p of v . As a corollary, we enlarge the set of known values of v for which there exists a dihedral SQS ( v ) . [ABSTRACT FROM AUTHOR]

Recently, linear codes with a few weights have been constructed and extensively studied. In this paper, for an odd prime p , we determine the complete weight enumerator of two classes of p -ary linear codes constructed from defining set. Our results show that the codes have at most seven weights and may have applications in secret sharing schemes. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex set V , minimum degree δ and edge-connectivity λ . If α is a real number, then the zeroth-order general Randić index is defined by ∑ x ∈ V deg α ( x ) , where deg ( x ) denotes the degree of the vertex x . A graph is maximally edge-connected if λ = δ . In this paper, we present sufficient conditions for connected graphs (resp. connected triangle-free graphs) to be maximally edge-connected in terms of the zeroth-order general Randić index, the order and the minimum degree when α ∈ ( − ∞ , 0 ) or α ∈ ( 1 , 2 ] (resp. α ∈ [ − 1 , 0 ) ∪ ( 1 , 2 ] ). [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a graph with no isolated vertex. A subset of vertices S is a total dominating set if every vertex of G is adjacent to some vertex of S . For some α with 0 < α ≤ 1 , a total dominating set S in G is an α -total dominating set if for every vertex v ∈ V ∖ S , | N ( v ) ⋂ S | ⩾ α | N ( v ) | . The α -total domination number of G , denoted by γ α t ( G ) , is the minimum cardinality of an α -total dominating set of G . In Henning and Rad (2012), Henning and Rad posed the following question: Let G be a connected cubic graph with order n . Is it true that γ α t ( G ) ≤ 3 n 4 for 2 3 < α ≤ 1 ? In this paper, we give a positive answer toward this question. Furthermore, we give a characterization on cubic graphs attaining the bound for the α -total domination number. [ABSTRACT FROM AUTHOR]

In their celebrated paper, Erdos et al. (1961) posed the following question. Let F be a family of k -element subsets of an n -element set satisfying the condition that | F ∩ G | ≥ ℓ holds for any two members of F where ℓ ≤ k are fixed positive integers. What is the maximum size | F | of such a family? They gave a complete solution for the case ℓ = 1 : the largest family is the one consisting of all k -element subsets containing a fixed element of the underlying set. (One has to suppose 2 k ≤ n , otherwise the problem is trivial.) They also proved that the best construction for arbitrary ℓ is the family consisting of all k - element subsets containing a fixed ℓ -element subset, but only for large n ’s. They also gave an example showing that this statement is not true for small n ’s. Later Frankl gave a construction for the general case that he believed to be the best. Frankl, Wilson and Füredi made serious progress towards the proof of this conjecture, but the complete solution was not achieved until 1996 when the surprising news came: Rudolf Ahlswede and Levon Khachatrian have found the proof. They invented the expressive name: Complete Intersection Theorem. We will show some of the consequences of this deep and important theorem. [ABSTRACT FROM AUTHOR]

Let d 1 , d 2 , … , d k be k non-negative integers. A graph G is ( d 1 , d 2 , … , d k ) -colorable, if the vertex set of G can be partitioned into subsets V 1 , V 2 , … , V k such that the subgraph G [ V i ] induced by V i has maximum degree at most d i for i = 1 , 2 , … , k . Borodin, Montassier and Raspaud asked: Is every planar graph without adjacent cycles of length at most five 3-colorable, i.e., ( 0 , 0 , 0 ) -colorable? This problem has now been answered negatively by Cohen-Addad et al. who successfully constructed a non-3-colorable planar graph with neither 4-cycles nor 5-cycles. Is every planar graph without adjacent cycles of length at most five ( 1 , 0 , 0 ) -colorable? To this new problem, this paper proves that every planar graph without adjacent cycles of length at most five is ( 1 , 1 , 0 ) -colorable. [ABSTRACT FROM AUTHOR]

A 3-way ( v , k , t ) trade of volume m consists of three disjoint collections T 1 , T 2 and T 3 , each of m blocks of size k , such that for every t -subset of v -set V , the number of blocks containing this t -subset is the same in each T i ( 1 ≤ i ≤ 3 ) . If any t -subset of found(T) occurs at most once in T 1 ( T j , j ≥ 2 ) , then T is called 3-way ( v , k , t ) Steiner trade. In this paper the spectrum (that is, the set of allowable volumes) of 3-way ( v , k , t ) Steiner trades is discussed. Here it is shown that the volume of a 3-way ( v , k , 2 ) Steiner trade is at least 3 ( k − 1 ) for k ≠ 4 . Also we show how to construct a 3-way ( v , k , 2 ) Steiner trade of volume m when m ≥ 12 ( k − 1 ) for k ≥ 15 , or m is multiple of three and 3 ( k − 1 ) ≤ m ≤ 12 ( k − 1 ) . [ABSTRACT FROM AUTHOR]