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A hypergraph ℋ is an (n, m)-hypergraph if it contains n vertices and m hyperedges, where n ≥ 1 and m ≥ 0 are two integers. Let k be a positive integer and let L be a set of nonnegative integers. A hypergraph ℋ is k-uniform if all its hyperedges have the same size k, and ℋ is L-intersecting if the number of common vertices of every two hyperedges belongs to L. In this paper, we propose and investigate the problem of estimating the maximum k among all k-uniform L-intersecting (n, m)-hypergraphs for fixed n, m and L. We will provide some tight upper and lower bounds on k in terms of n, m and L. [ABSTRACT FROM AUTHOR]

Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ. [ABSTRACT FROM AUTHOR]

This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let $$\left\{ {X_a } \right\}_{a \in Z_ + }$$ be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1, 2, ...}. Let T ⊂ S be an arbitrary finite subset of S. Denote by Ln the length of the longest run of consecutive i’s for i ∈ T, that occurs in the sequence X1, ..., Xn. In this paper, we obtain a limit law and a week version of an Erdös-Rényi type law for Ln. A large deviation result of Ln is also discussed. [ABSTRACT FROM AUTHOR]

Let G be a finite group, Irr( G) be the set of nonlinear irreducible characters of G and cd( G) the set of degrees of the characters in Irr( G). A group G is said to be a D -group if |cd( G)| = |Irr( G)| − 2. In this paper, we give a complete classification of solvable D -groups. [ABSTRACT FROM AUTHOR]

Let X denote a compact metric space with distance d and F : X × ℝ → X or F : X → X denote a C -flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C -flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Hyers-Ulam-Rassias stability of isometric homomorphisms in proper CQ*-algebras for the following Cauchy-Jensen additive mapping: The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias’ stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. This is applied to investigate isometric isomorphisms between proper CQ*-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous. [ABSTRACT FROM AUTHOR]

Let { X, X; n ≥ 0} be a sequence of independent and identically distributed random variables with E X = 0, and assume that E X I(| X| ≤ x) is slowly varying as x→∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series $$\sum\nolimits_{n = 0}^\infty {{\beta ^n}{X_n}\left( {0 < \beta < 1} \right)} $$ is obtained, under some minimal conditions. [ABSTRACT FROM AUTHOR]

We first determine in this paper the structure of the generalized Fitting subgroup F* ( G) of the finite groups G all of whose defect groups (of blocks) are conjugate under the automorphism group Aut( G) to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/O( L) has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O( L) is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity. [ABSTRACT FROM AUTHOR]

A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color. In this paper, it is proved that if G is a planar graph with Δ( G) ≥ 7 and without chordal 7-cycles, then G has a (Δ( G)+1)-total-coloring. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence, uniqueness and asymptotic behavior of steady states to a class of Schrödinger-Poisson-Slater System. [ABSTRACT FROM AUTHOR]

Let Π = B/ B/ ··· / B be any set partition of [ n] = {1, 2,..., n} satisfying that entries are increasing in each block and blocks are arranged in increasing order of their first entries. Then Callan defined the flattened Π to be the permutation of [ n] obtained by erasing the divers between its blocks, and Callan also enumerated the number of set partitions of [ n] whose flattening avoids a single 3-letter pattern. Mansour posed the question of counting set partitions of [ n] whose flattening avoids a pattern of length 4. In this paper, we present the number of set partitions of [ n] whose flattening avoids one of the patterns: 1234, 1243, 1324, 1342, 1423, 1432, 3142 and 4132. [ABSTRACT FROM AUTHOR]

Suppose A is a unital C*-algebra and r > 1. In this paper, we define a unital C*-algebra C*( A, r) and a completely bounded unital homomorphism α: A → C*( A, r) with the property that C*( A, r) = C*( α( A)) and, for every unital C*-algebra B and every unital completely bounded homomorphism φ: A → B, there is a (unique) unital *-homomorphism π: C*( A, r) → B such that φ = π ◦ α. We prove that, if A is generated by a normal set { t: λ ∈ Λ}, then C*( A, r) is generated by the set { α( t): λ ∈ Λ}. By proving an equation of the norms of elements in a dense subset of C*( A, r) we obtain that, if B is a unital C*-algebra that can be embedded into A, then C*( B, r) can be naturally embedded into C*( A, r). We give characterizations of C*( A, r) for some special situations and we conclude that C*( A, r) will be 'nice' when dim( A) ≤ 2 and 'quite complicated' when dim( A) ≥ 3. We give a characterization of the relation between K-groups of A and K-groups of C*( A, r). We also define and study some analogous of C*( A, r). [ABSTRACT FROM AUTHOR]

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined. [ABSTRACT FROM AUTHOR]

Let G be a finite centerless group, let π( G) be the set of prime divisors of the order of G, and let n( G) be the number of Sylow p-subgroups of G, that is, n( G) = |Syl( G)|. Set NS( G):= { n( G)| p ∈ π( G)}. In this paper, we are investigating whether L( r) is determined up to isomorphism by NS( L( r)) when r is prime. [ABSTRACT FROM AUTHOR]

This paper considers the problem of n-widths of a Sobolev function class Ω determined by P( D) = D Π( D − t I) in Orlicz spaces. The relationship between the extreme value problem and width theory is revealed by using the methods of functional analysis. Particularly, as σ = 0 or σ = 1, the exact values of Kolmogorov's widths, Gelfand's widths, and linear widths are obtained respectively, and the related extremal subspaces and optimal linear operators are given. [ABSTRACT FROM AUTHOR]

This paper studies the quasisymmetric mappings on Moran sets. We introduce a generalized form of weak quasisymmetry and prove that, on Moran set satisfying the small gap condition, a generalized weakly quasisymmetric mapping is quasisymmetric. We further give a criterion for the quasisymmetry of mappings between Moran sets with some regular structure. [ABSTRACT FROM AUTHOR]

The first part of this article presents invariance criteria for a stochastic differential equation whose state evolution is constrained by time-dependent security tubes. The key results of this section are derived by considering an equivalent problem where the square of distance function represents a viscosity solution to an adequately defined partial differential equation. The second part of the paper analyzes the broader context when solutions are constrained by more general time-dependent convex domains. The approach relies on forward stochastic variational inequalities with oblique reflection, the generalized subgradients acting as a reacting process that operates only when the solution reaches the boundary of the domain. [ABSTRACT FROM AUTHOR]

A finite group is inseparable, it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect product decompositions. This paper investigates two classes of finite inseparable p-groups and, for p ≥ 3, establishes a necessary and sufficient condition for inseparability. [ABSTRACT FROM AUTHOR]

In this paper, we show that (1) for each QFS-domain L, L is an ωQFS-domain iff L has a countable base for the Scott topology; (2) the Scott-continuous retracts of QFS-domains are QFS-domains; (3) for a quasicontinuous domain L, L is Lawson compact iff L is a finitely generated upper set and for any x, x ∈ L and finite G, G ⊆ L with G ≪ x, G ≪ x, there is a finite subset F ⊆ L such that ↑ x∩ ↑ x ⊆↑ F ⊆↑ G∩ ↑ G; (4) L is a QFS-domain iff L is a quasicontinuous domain and given any finitely many pairs {( F, x): F is finite, x ∈ L with F ≪ x, 1 ≤ i ≤ n}, there is a quasi-finitely separating function δ on L such that F ≪ δ( x) ≪ x. [ABSTRACT FROM AUTHOR]

A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set [ k] = {1, 2, ..., k}. Let f( u) denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E( G), f( u) ≠ f( v). By χ″( G), we denote the smallest value k in such a coloring of G. Pilśniak and Woźniak conjectured that χ″( G) ≤ Δ( G)+3 for any simple graph with maximum degree Δ( G). In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7. [ABSTRACT FROM AUTHOR]

In this paper, we give a criterion of whether there are non-wandering expanding maps on a given branched 1-manifold. As an application, we give an example of a dynamic on a 3-manifold having non-zero first Betti number, the non-wander sets of which are two Smale-Williams solenoids. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear and nonautonomous impulsive stochastic functional differential equation is considered. By establishing a nonautonomous L-operator impulsive delay inequality and using the properties of ρ-cone and stochastic analysis technique, we obtain the p-attracting set and p-invariant set of the impulsive stochastic functional differential equation. An example is also discussed to illustrate the efficiency of the obtained results. [ABSTRACT FROM AUTHOR]

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex ν ∈ V ( G) at most f( ν) times. The f-core of G is the subgraph of G induced by the vertices ν of degree $d(v) = f(v)\max _{v \in V(G)} \{ \left\lceil {d(v)/f(v)} \right\rceil \} $. In this paper, we find some necessary conditions for a simple graph, whose f-core has maximum degree two, to be of class 2 for f-colorings. [ABSTRACT FROM AUTHOR]

A proper [ h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [ h] = {1, 2, ..., h}. Let w( u) denote the sum of the color on a vertex u and colors on all the edges incident to u. For each edge u ∈ E( G), if w( u) ≠ w( v), then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [ h]-total coloring of G. By tndi ( G), we denote the smallest value h in such a coloring of G. In this paper, we obtain that G is a graph with at least two vertices, if mad( G) < 3, then tndi ( G) ≤ k + 2 where k = max{Δ( G), 5}. It partially confirms the conjecture proposed by Pilśniak and Woźniak. [ABSTRACT FROM AUTHOR]

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by τ( G), is the minimum cardinality of a clique-transversal set in G. In this paper, we first present a lower bound on τ( G) and characterize the extremal graphs achieving the lower bound for a connected (claw, K)-free 4-regular graph G. Furthermore, we show that for any 2-connected (claw, K)-free 4-regular graph G of order n, its clique-transversal number equals to ⌈ n/3⌉. [ABSTRACT FROM AUTHOR]

Let G ( n, k, t) be a set of graphs with n vertices, k cut edges and t cut vertices. In this paper, we classify these graphs in G ( n, k, t) according to cut vertices, and characterize the extremal graphs with the largest spectral radius in G ( n, k, t). [ABSTRACT FROM AUTHOR]

Let G = ( V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted γ( G), is the smallest cardinality of a restrained dominating set of G. In this paper, we show that if G is a graph of order n ≥ 4, then $$\gamma _r \left( G \right)\gamma _r \left( {\bar G} \right) \leqslant 2n$$. We also characterize the graphs achieving the upper bound. [ABSTRACT FROM AUTHOR]

To decide when a graph is Gromov hyperbolic is, in general, a very hard problem. In this paper, we solve this problem for the set of short graphs (in an informal way, a graph G is r-short if the shortcuts in the cycles of G have length less than r): an r-short graph G is hyperbolic if and only if S( G) is finite, where S( G):= sup{ L( C): C is an R-isometric cycle in G} and we say that a cycle C is R-isometric if d( x, y) ≤ d( x, y) + R for every x, y ∈ C. [ABSTRACT FROM AUTHOR]

An antimagic labeling of a graph with q edges is a bijection from the set of edges to the set of positive integers {1, 2, ..., q} such that all vertex weights are pairwise distinct, where the vertex weight of a vertex is the sum of the labels of all edges incident with that vertex. A graph is antimagic if it has an antimagic labeling. In this paper, we provide antimagic labelings for a family of generalized pyramid graphs. [ABSTRACT FROM AUTHOR]

Let G be a graph, and k ≥ 2 be a positive integer. A graph G is fractional independentset-deletable k-factor-critical (in short, fractional ID- k-factor-critical), if G-I has a fractional k-factor for every independent set I of G. The binding number bind( G) of a graph G is defined as $$bind(G) = min\left\{ {\frac{{\left| {N_G (X)} \right|}} {{\left| X \right|}}:\not 0 \ne X \subseteq V(G),N_G (X) \ne V(G)} \right\}.$$ In this paper, it is proved that a graph G is fractional ID- k-factor-critical if n ≥ 6 k − 9 and bind( G) $$> \frac{{(3k - 1)(n - 1)}} {{kn - 2k + 2}}$$ . [ABSTRACT FROM AUTHOR]

The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs. [ABSTRACT FROM AUTHOR]

Difference systems of sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zν relative to subgroup of order k, where ν = km is a composite number. As applications, we obtain some new optimal DSSs. [ABSTRACT FROM AUTHOR]

A subset system Z assigns to each partially ordered set P a certain collection Z( P) of subsets. In this paper, a new kind of subset systems called directable subset systems is introduced. For a directable subset system Ƶ, the concepts of FƵ-way-below relation and FƵ-domain are introduced. The well-known Scott topology is naturally generalized to the Ƶ-level and the resulting topology is called FƵ-Scott topology, and the continuous functions with respect to this topology are characterized by preserving the suprema of directed Ƶ-sets. Then, we mainly consider a generalization of the cartesian closedness of the categories DCPO of directed complete posets, BF of bifinite domains and FS of FS-domains to the Ƶ-level. Corresponding to them, it is proved that, for a suitable subset system Ƶ, the categories FƵCPO of Ƶ-complete posets, FSFƵ of finitely separated FƵ-domains and BFFƵ of bifinite FƵ-domains are all cartesian closed. Some examples of these categories are given. [ABSTRACT FROM AUTHOR]

Let X, Y be two real Banach spaces and ɛ ≥ 0. A map f: X → Y is said to be a standard ɛ-isometry if |‖ f( x) − f( y)‖ − ‖ x − y‖| ≤ ɛ for all x, y ∈ X and with f(0) = 0. We say that a pair of Banach spaces ( X, Y) is stable if there exists γ > 0 such that, for every such ɛ and every standard ɛ-isometry f: X → Y, there is a bounded linear operator $T:L(f) \equiv \overline {span} f(X) \to X$ so that ‖ Tf( x)t- x‖ ≤ γɛ for all x ∈ X. X( Y) is said to be universally left-stable if ( X, Y) is always stable for every Y ( X). In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w*-closed subspace of ℓ∞(Γ) for some set Γ, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant. [ABSTRACT FROM AUTHOR]

In this paper, we give some decompositions of triples of $\mathbb{Z}_{p^n } $ or $\mathbb{Z}_{3p^n } $ into cyclic triple systems. New constructions of difference families are given. Some infinite classes of simple cyclic triple systems are obtained from these decompositions. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new class of finite groups whose common divisor graphs are complete graphs, while there is no prime dividing all the nontrivial degrees. [ABSTRACT FROM AUTHOR]

In regularized kernel methods, the solution of a learning problem is found by minimizing a functional consisting of a empirical risk and a regularization term. In this paper, we study the existence of optimal solution of multi-kernel regularization learning. First, we ameliorate a previous conclusion about this problem given by Micchelli and Pontil, and prove that the optimal solution exists whenever the kernel set is a compact set. Second, we consider this problem for Gaussian kernels with variance σ ∈ (0,∞), and give some conditions under which the optimal solution exists. [ABSTRACT FROM AUTHOR]

In this paper, we consider a discrete version of Aleksandrov's projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice's y-coordinates' absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov's projection theorem which was proposed by Gardner, Gronchi and Zong in 2005. [ABSTRACT FROM AUTHOR]

A subset S of vertices of a graph G with no isolated vertex is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex in V ( G) − S is also adjacent to a vertex in V ( G) − S. The total restrained domination number of G is the minimum cardinality of a total restrained dominating set of G. In this paper we initiate the study of total restrained bondage in graphs. The total restrained bondage number in a graph G with no isolated vertex, is the minimum cardinality of a subset of edges E such that G - E has no isolated vertex and the total restrained domination number of G - E is greater than the total restrained domination number of G. We obtain several properties, exact values and bounds for the total restrained bondage number of a graph. [ABSTRACT FROM AUTHOR]

In this paper, we apply a variant of the famous Mountain Pass Lemmas of Ambrosetti-Rabinowitz and Ambrosetti-Coti Zelati with (PSC) c type condition of Palais-Smale-Cerami to study the existence of new periodic solutions with a prescribed energy for symmetrical singular second order Hamiltonian conservative systems with weak force type potentials. [ABSTRACT FROM AUTHOR]

A weakly pandiagonal Latin square of order n over the number set {0, 1, ..., n − 1} is a Latin square having the property that the sum of the n numbers in each of 2 n diagonals is the same. In this paper, we shall prove that a pair of orthogonal weakly pandiagonal Latin squares of order n exists if and only if n ≡ 0, 1,3 (mod 4) and n ≠ 3. [ABSTRACT FROM AUTHOR]

A labeling f of a graph G is a bijection from its edge set E( G) to the set {1, 2, ..., | E( G)|}, which is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. A graph G is antimagic if G has an f which is antimagic. Hartsfield and Ringel conjectured in 1990 that every connected graph other than K2 is antimagic. In this paper, we show that if G is an n-vertex graph with minimum degree at least r, and G2 is an m-vertex graph with maximum degree at most 2 r − 1 ( m ≥ n), then G ∀ G is antimagic. [ABSTRACT FROM AUTHOR]

In this paper, we prove a quantitative version of the statement that every nonempty finite subset of ℕ is a set of quadratic residues for infinitely many primes of the form [ n] with 1 ≤ c ≤ 243/205. Correspondingly, we can obtain a similar result for the case of quadratic non-residues under reasonable assumptions. These results generalize the previous ones obtained by Wright in certain aspects. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the definition of ( m, n)-regularity in Γ-semigroups. We investigate and characterize the 2-regular class of Γ-semigroups using Green's relations. Extending and generalizing the Croisot's Theory of Decomposition for Γ-semigroups, we introduce and study the absorbent and regular absorbent Γ-semigroups. We approach this problem by examining quasi-ideals using Green's relations. [ABSTRACT FROM AUTHOR]

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number f( G) is the smallest number m such that for every distribution of m pebbles and every vertex v, a pebble can be moved to v. A graph G is said to have the 2- pebbling property if for any distribution with more than 2 f( G)- q pebbles, where q is the number of vertices with at least one pebble, it is possible, using pebbling moves, to get two pebbles to any vertex. Snevily conjectured that G( s, t) has the 2-pebbling property, where G( s, t) is a bipartite graph with partite sets of size s and t ( s ≥ t). Similarly, the ℓ-pebbling number f( G) is the smallest number m such that for every distribution of m pebbles and every vertex v, ℓ pebbles can be moved to v. Herscovici et al. conjectured that f( G) ≤ 1.5 n + 8 ℓ −6 for the graph G with diameter 3, where n = | V ( G)|. In this paper, we prove that if s ≥ 15 and G( s, t) has minimum degree at least $\left\lceil {\tfrac{{s + 1}} {2}} \right\rceil $ , then f( G( s, t)) = s + t, G( s, t) has the 2-pebbling property and f( G( s, t)) ≤ s + t + 8( ℓ − 1). In other words, we extend a result due to Czygrinow and Hurlbert, and show that the above Snevily conjecture and Herscovici et al. conjecture are true for G( s, t) with s ≥ 15 and minimum degree at least $\left\lceil {\tfrac{{s + 1}} {2}} \right\rceil $ . [ABSTRACT FROM AUTHOR]

In this paper, we prove that some volume-preserving almost Anosov systems are ergodic if they are essentially accessible. The key idea is that there are stable and unstable manifolds with uniform size on the orbits of the hyperbolic points for these systems. [ABSTRACT FROM AUTHOR]

Let M be a closed smooth manifold M, and let f: M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f. We show that if f has the C-stably average shadowing property on Λ, then Λ admits a dominated splitting. [ABSTRACT FROM AUTHOR]

For a simple digraph G, let β( G) be the size of the smallest subset X ⊆ E( G) such that G−X has no directed cycles, and let γ( G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that β( G) ≤ 0.3819 γ( G) if G is a 4-free digraph, and β( G) ≤ 0.2679 γ( G) if G is a 5-free digraph. These improve the results of Sullivan in 2008. [ABSTRACT FROM AUTHOR]

The induced path number ρ( G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. Broere et al. proved that if G is a graph of order n, then $$\sqrt n \leqslant \rho \left( G \right) + \rho \left( {\bar G} \right) \leqslant \left\lceil {\tfrac{{3n}} {2}} \right\rceil$$. In this paper, we characterize the graphs G for which $$\rho \left( G \right) + \rho \left( {\bar G} \right) = \left\lceil {\tfrac{{3n}} {2}} \right\rceil$$, improve the lower bound on $$\rho \left( G \right) + \rho \left( {\bar G} \right)$$ by one when n is the square of an odd integer, and determine a best possible upper bound for $$\rho \left( G \right) + \rho \left( {\bar G} \right)$$ when neither G nor $$\bar G$$ has isolated vertices. [ABSTRACT FROM AUTHOR]