On the eccentric distance sum of trees and unicyclic graphs

Let G be a simple connected graph with the vertex set V(G). The eccentric distance sum of G is defined as ξd(G)=_∑v∈V(G)ε(v)DG(v), where ε(v) is the eccentricity of the vertex v and DG(v)=_∑u∈V(G)d(u,v) is the sum of all distances from the vertex v. In this paper we characterize the extremal unicyclic graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum. In addition, we characterize the extremal trees with given diameter and minimal eccentric distance sum.     

Domination with exponential decay

Let G be a graph and S⊆V(G). For each vertex u∈S and for each v∈V(G)−S, we define to be the length of a shortest path in 〈V(G)−(S−{u})〉 if such a path exists, and ∞ otherwise. Let v∈V(G). We define if v⁄∈S, and w_S(v)=2 if v∈S. If, for each v∈V(G), we have w_S(v)≥1, then S is an exponential dominating set. The smallest cardinality of an exponential dominating set is the exponential domination number, γ_e(G). In this paper, we prove: (i) that if G is a connected graph of diameter d, then γ_e(G)≥(d+2)/4, and, (ii) that if G is a connected graph of order n, then .     

A note on Roman domination in graphs

Let G=(V,E) be a simple graph. A subset S⊆V is a dominating set of G, if for any vertex u∈V-S, there exists a vertex v∈S such that uv∈E. The domination number of G, γ(G), equals the minimum cardinality of a dominating set. A Roman dominating function on graph G=(V,E) is a function f:V→{0,1,2} satisfying the condition that every vertex v for which f(v)=0 is adjacent to at least one vertex u for which f(u)=2. The weight of a Roman dominating function is the value f(V)=∑_v∈Vf(v). The Roman domination number of a graph G, denoted by γ_R(G), equals the minimum weight of a Roman dominating function on G. In this paper, for any integer k(2?k?γ(G)), we give a characterization of graphs for which γ_R(G)=γ(G)+k, which settles an open problem in [E.J. Cockayne, P.M. Dreyer Jr, S.M. Hedetniemi et al. On Roman domination in graphs, Discrete Math. 278 (2004) 11-22].     

Unique response Roman domination in graphs

A function f:V(G)→{0,1,2} is a Roman dominating function if every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. A function f:V(G)→{0,1,2} with the ordered partition (V₀,V₁,V₂) of V(G), where V_i={v∈V(G)∣f(v)=i} for i=0,1,2, is a unique response Roman function if x∈V₀ implies |N(x)∩V₂|≤1 and x∈V₁∪V₂ implies that |N(x)∩V₂|=0. A function f:V(G)→{0,1,2} is a unique response Roman dominating function if it is a unique response Roman function and a Roman dominating function. The unique response Roman domination number of G, denoted by u_R(G), is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination number of graphs and present bounds for this parameter.     

Variable neighborhood search for extremal graphs. 23. On the Randi? index and the chromatic number

Let G=(V,E) be a simple graph with vertex degrees d₁,d₂,…,d_n. The Randi? index R(G) is equal to the sum over all edges (i,j)∈E of weights . We prove several conjectures, obtained by the system AutoGraphiX, relating R(G) and the chromatic number χ(G). The main result is χ(G)≤2R(G). To prove it, we also show that if v∈V is a vertex of minimum degree δ of G, G−v the graph obtained from G by deleting v and all incident edges, and Δ the maximum degree of G, then .     

Perturbed sampling formulas and local reconstruction in shift invariant spaces

Let V? be a closed subspace of L₂(R) generated from the integer shifts of a continuous function ? with a certain decay at infinity and a non-vanishing property for the function Φ^†(γ)=_∑n∈Z?(n)e−inγ on [−π,π]. In this paper we determine a positive number δ? so that the set {n+δn}_n∈Z is a set of stable sampling for the space V? for any selection of the elements δn within the ranges ±δ?. We demonstrate the resulting sampling formula (called perturbation formula) for functions f∈V? and also we establish a finite reconstruction formula approximating f on bounded intervals. We compute the corresponding error and we provide estimates for the jitter error as well.     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

Moderate deviations and Erdös-Rényi-Shepp laws for weighted sums

Let X₀,X₁,… be i.i.d. random variables with E(X₀)=0, E(X²₀)=1 and E(exp{tX₀})<∞ for any |t|<t₀. We prove that the weighted sums V(n)=∑_j=0^∞a_j(n)X_j, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums.     

Local approximations for maximum partial subgraph problem

We deal with MAXH₀-FREE PARTIAL SUBGRAPH. We mainly prove that 3-locally optimum solutions achieve approximation ratio (δ₀+1)/(B+2+ν₀), where B=max_v∈Vd_G(v), δ₀=min_v∈V(H0)d_H0(v) and ν₀=(|V(H₀)|+1)/δ₀. Next, we show that this ratio rises up to 3/(B+1) when H₀=K₃. Finally, we provide hardness results for MAXK₃-FREE PARTIAL SUBGRAPH.     

Balanced decomposition of a vertex-colored graph

A balanced vertex-coloring of a graph G is a function c from V(G) to {−1,0,1} such that ∑{c(v):v∈V(G)}=0. A subset U of V(G) is called a balanced set if U induces a connected subgraph and ∑{c(v):v∈U}=0. A decomposition V(G)=V₁∪?∪V_r is called a balanced decomposition if V_i is a balanced set for 1≤i≤r.In this paper, the balanced decomposition number f(G) of G is introduced; f(G) is the smallest integer s such that for any balanced vertex-coloring c of G, there exists a balanced decomposition V(G)=V₁∪?∪V_r with |V_i|≤s for 1≤i≤r. Balanced decomposition numbers of some basic families of graphs such as complete graphs, trees, complete bipartite graphs, cycles, 2-connected graphs are studied.     

An Example of the Effect of Rescaling of the Reflection Coefficient on the Scattering Potential for the One-Dimensional Schrodinger Equation

Let us consider the one-dimensional Schrödinger equation with the scattering potential V(x)=2δ(x) and corresponding reflection coefficient b(k)=?i/(k + i). The potential satisfies a theorem of Deift and Trubowitz which states that non-negative measurable potentials V(x) satisfying a certain range condition have reflection coefficients b(k) such that b(0)=?1. We rescale the reflection coefficient for V(x)=2δ(x) by writing b(k)=?iv/(k + 1) where 0<v<1. It is shown how V(x) changes with v, through the use of the Gelfand-Levitan equation. This example illustrates how sensitive the potential is to rescaling of the reflection coefficient. In particular, the rescaling leads to a negative portion of V(x), as is expected from the Deift-Trubowitz theorem. The example of this paper will be used in a later paper to illustrate the nature of bounds on potentials obtained through the use of variational principles.     

Global Strichartz estimates for the wave equation with time-periodic potentials

We obtain global Strichartz estimates for the solutions u of the wave equation for time-periodic potentials V(t,x) with compact support with respect to x. Our analysis is based on the analytic properties of the cut-off resolvent Rχ(z)=χ(U⁻¹(T)−zI)ψ₁, where U(T)=U(T,0) is the monodromy operator and T>0 the period of V(t,x). We show that if Rχ(z) has no poles z∈C, |z|?1, then for n?3, odd, we have a exponential decal of local energy. For n?2, even, we obtain also an uniform decay of local energy assuming that Rχ(z) has no poles z∈C, |z|?1, and Rχ(z) remains bounded for z in a small neighborhood of 0.     

Greedy F-colorings of graphs

Let G=(V,E) be a connected graph. For a symmetric, integer-valued function δ on V×V, where K is an integer constant, N₀ is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping by F(u,v,m)=δ(u,v)+m−K. A coloring c of G is an F-coloring if F(u,v,|c(u)−c(v)|)?0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F(G) is the minimum value among all F-colorings of G. For an ordering of the vertices of G, a greedy F-coloring c of s is defined by (1) c(v₁)=1 and (2) for each i with 1?i<n, c(v_i+1) is the smallest positive integer p such that F(v_j,v_i+1,|c(v_j)−p|)?0, for each j with 1?j?i. The greedy F-chromatic number gF(s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF(G)=min{gF(s)} over all orderings s of V. The Grundy F-chromatic number is GF(G)=max{gF(s)} over all orderings s of V. It is shown that gF(G)=F(G) for every graph G and every F-coloring defined on G. The parameters gF(G) and GF(G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ(u,v) is the distance between u and v and .     

Precoloring extension for 2-connected graphs with maximum degree three

Let G=G(V,E) be a simple graph, L a list assignment with |L(v)|=Δ(G)∀v∈V and W⊆V an independent subset of the vertex set. Define to be the minimum distance between two vertices of W. In this paper it is shown that if G is 2-connected with Δ(G)=3 and G is not K₄ then every precoloring of W is extendable to a proper list coloring of G provided that d(W)≥6. An example shows that the bound is sharp. This result completes the investigation of precoloring extensions for graphs with |L(v)|=Δ(G) for all v∈V where the precolored set W is an independent set.     

On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G and Π=(C₁,C₂,…,C_k) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c_Π(v):=(d(v,C₁),d(v,C₂),…,d(v,C_k)), where d(v,C_i)=min{d(v,x)|x∈C_i},1≤i≤k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ_L(G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ_L(KG(n,2))=n−1 for all n≥5. Then, we prove that χ_L(KG(n,k))≤n−1, when n≥k². Moreover, we present some bounds for the locating chromatic number of odd graphs.     

Coloring graphs from random lists of size 2

Let G=G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set C of size σ(n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring φ such that φ(v)∈L(v) for all v∈V(G). In particular, we show that if g is odd and σ(n)=ω(n^1/(2g−2)), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n→∞. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n≥g, there is a graph H=H(n,g) with bounded maximum degree and girth g, such that if σ(n)=o(n^1/(2g−2)), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n→∞. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size σ(n), exhibits a sharp threshold at σ(n)=2n.     

Near automorphisms of cycles

Let f be a permutation of V(G). Define δ_f(x,y)=|d_G(x,y)-d_G(f(x),f(y))| and δ_f(G)=∑δ_f(x,y) over all the unordered pairs {x,y} of distinct vertices of G. Let π(G) denote the smallest positive value of δ_f(G) among all the permutations f of V(G). The permutation f with δ_f(G)=π(G) is called a near automorphism of G. In this paper, we study the near automorphisms of cycles C_n and we prove that π(C_n)=4⌊n/2⌋-4, moreover, we obtain the set of near automorphisms of C_n.     

Eigenvalues and degree deviation in graphs

Let G be a graph with n vertices and m edges and let μ(G) = μ₁(G) ? ? ? μ_n(G) be the eigenvalues of its adjacency matrix. Set s(G)=∑_u∈V(G)∣d(u)-2m/n∣. We prove that

     

The source location problem with local 3-vertex-connectivity requirements

Let G=(V,E) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v∈V has an integer valued demand d(v)?0. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices with the minimum cardinality such that there are at least d(v) vertex-disjoint paths between S and each vertex v∈V-S. In this paper, we show that the problem with d(v)?3, v∈V can be solved in linear time. Moreover, we show that in the case where d(v)?4 for some vertex v∈V, the problem is NP-hard.     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.