Sigma chromatic number of graph coronas involving complete graphs

Let c : V(G) → be a coloring of the vertices in a graph G. For a vertex u in G, the color sum of u, denoted by σ(u), is the sum of the colors of the neighbors of u. The coloring c is called a sigma coloring of G if σ(u) ≠ σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). Given two simple, connected graphs G and H, the corona of G and H, denoted by G ⊙ H, is the graph obtained by taking one copy of G and |V(G)| copies of H and where the ith vertex of G is adjacent to every vertex of the ith copy of H. In this study, we will show that for a graph G with |V(G)| ≥ 2, and a complete graph Kn of order n, n ≤ σ(G ⊙ Kn ) ≤ max {σ(G), n}. In addition, let Pn and Cn denote a path and a cycle of order n respectively. If m, n ≥ 3, we will prove that σ(Km ⊙ Pn ) = 2 if and only if . If n is even, we show that σ(Km ⊙ Cn ) = 2 if and only if . Furthermore, in the case that n is odd, we show that σ(Km ⊙ Cn ) = 3 if and only if where H(r, s) denotes the number of lattice points in the convex hull of points on the plane determined by the integer parameters r and s

Twin chromatic indices of some graphs with maximum degree 3

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in k ) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by . In this paper, we determine the twin chromatic indices of circulant graphs , and some generalized Petersen graphs such as GP(3s, k), GP(m, 2), and GP(4s, l) where n ≥ 6 and n ≡ 0 (mod 4), s ≥ 1, k ≢ 0 (mod 3), m ≥ 3 and m {4, 5}, and l is odd. Moreover, we provide some sufficient conditions for a connected graph with maximum degree 3 to have twin chromatic index greater than 3

On the Sigma Value and Sigma Range of the Join of a Finite Number of Even Cycles of the Same Order

Let c be a vertex coloring of a simple; connected graph G that uses positive integers for colors. For a vertex v of G; the color sum of v is the sum of the colors of the neighbors of v. If no two adjacent vertices of G have the same color sum; then c is called a sigma coloring of G. The sigma chromatic number of G is the minimum number of colors required in a sigma coloring of G. Let max(c) be the largest color assigned to a vertex of G by a coloring c. The sigma value of G is the minimum value of max(c) over all sigma k−colorings c of G where k is the sigma chromatic number of G. On the other hand; the sigma range of G is the minimum value of max(c) over all sigma colorings c of G. In this paper; we determine the sigma value and the sigma range of the join of a finite number of even cycles of the same order

On twin edge colorings in m-ary trees

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from ℤk and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in ℤk) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by χ′t(G). In this paper, we study the twin edge colorings in m-ary trees for m ≥ 2; in particular, the twin chromatic indexes of full m-ary trees that are not stars, r-regular trees for even r ≥ 2, and generalized star graphs that are not paths nor stars are completely determined. Moreover, our results confirm the conjecture that χ′t(G)≤Δ(G)+2 for every connected graph G (except C5) of order at least 3, for all trees of order at least 3

Sigma Coloring and Edge Deletions

A vertex coloring c : V(G) → N of a non-trivial graph G is called a sigma coloring if σ(u) is not equal to σ(v) for any pair of adjacent vertices u and v. Here, σ(x) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the fewest number of colors needed to construct a sigma coloring of G. In this paper, we consider the sigma chromatic number of graphs obtained by deleting one or more of its edges. In particular, we study the difference σ(G)−σ(G−e) in general as well as in restricted scenarios; here, G−e is the graph obtained by deleting an edge e from G. Furthermore, we study the sigma chromatic number of graphs obtained via multiple edge deletions in complete graphs by considering the complements of paths and cycles

On the Sigma Chromatic Number of the Zero-Divisor Graphs of the Ring of Integers Modulo n

The zero-divisor graph of a commutative ring R with unity is the graph Γ(R) whose vertex set is the set of nonzero zero divisors of R; where two vertices are adjacent if and only if their product in R is zero. A vertex coloring c : V (G) → Bbb N of a non-trivial connected graph G is called a sigma coloring if σ(u) = σ(ν) for any pair of adjacent vertices u and v. Here; σ(χ) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G; denoted by σ(G); is defined as the least number of colors needed to construct a sigma coloring of G. In this paper; we analyze the structure of the zero-divisor graph of rings Bbb Zn; where n = pn11 P2n2 ...Pmnm; where m,ni,n2; ...,nm are positive integers and p1,p2; ...,pm are distinct primes. The analysis is carried out by partitioning the vertex set of such zero-divisor graphs and analyzing the adjacencies; cardinality; and the degree of the vertices in each set of the partition. Using these properties; we determine the sigma chromatic number of these zero-divisor graphs

The sigma chromatic number of the Sierpinski gasket graphs and the Hanoi graphs

A vertex coloring c : V(G) → of a non-trivial connected graph G is called a sigma coloring if σ(u) ≠ σ(v) for any pair of adjacent vertices u and v. Here, σ(x) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the fewest number of colors needed to construct a sigma coloring of G. In this paper, we determine the sigma chromatic numbers of the Sierpiński gasket graphs and the Hanoi graphs. Moreover, we prove the uniqueness of the sigma coloring for Sierpiński gasket graphs

Ancient Dna Sequence Revealed By Error-correcting Codes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)A previously described DNA sequence generator algorithm (DNA-SGA) using error-correcting codes has been employed as a computational tool to address the evolutionary pathway of the genetic code. The code-generated sequence alignment demonstrated that a residue mutation revealed by the code can be found in the same position in sequences of distantly related taxa. Furthermore, the code-generated sequences do not promote amino acid changes in the deviant genomes through codon reassignment. A Bayesian evolutionary analysis of both code-generated and homologous sequences of the Arabidopsis thaliana malate dehydrogenase gene indicates an approximately 1 MYA divergence time from the MDH code-generated sequence node to its paralogous sequences. The DNA-SGA helps to determine the plesiomorphic state of DNA sequences because a single nucleotide alteration often occurs in distantly related taxa and can be found in the alternative codon patterns of noncanonical genetic codes. As a consequence, the algorithm may reveal an earlier stage of the evolution of the standard code.5Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [2008/52067-3, 2008/04992-0, 2011/00417-3]CNPq [303059/2010-9, 503891/2011-8

Using continuous measurement to protect a universal set of quantum gates within a perturbed decoherence-free subspace

We consider a universal set of quantum gates encoded within a perturbed decoherence-free subspace of four physical qubits. Using second-order perturbation theory and a measuring device modeled by an infinite set of harmonic oscillators, simply coupled to the system, we show that continuous observation of the coupling agent induces inhibition of the decoherence due to spurious perturbations. We thus advance the idea of protecting or even creating a decoherence-free subspace for processing quantum information.Comment: 7 pages, 1 figure. To be published in Journal of Physics A: Mathematical and Genera

Alternative fidelity measure for quantum states

We propose an alternative fidelity measure (namely, a measure of the degree of similarity) between quantum states and benchmark it against a number of properties of the standard Uhlmann-Jozsa fidelity. This measure is a simple function of the linear entropy and the Hilbert-Schmidt inner product between the given states and is thus, in comparison, not as computationally demanding. It also features several remarkable properties such as being jointly concave and satisfying all of "Jozsa's axioms". The trade-off, however, is that it is supermultiplicative and does not behave monotonically under quantum operations. In addition, new metrics for the space of density matrices are identified and the joint concavity of the Uhlmann-Jozsa fidelity for qubit states is established.Comment: 12 pages, 3 figures. v2 includes minor changes, new references and new numerical results (Sec. IV