The k-dimensional Weisfeiler–Leman algorithm is a powerful tool in graph isomorphism testing. For an input G, the determines canonical coloring of s-tuples vertices G for each s between 1 and k. We say that numerical parameter k-WL-invariant if it determined by tuple color. As application Dvo?ák’s result on k-WL-invariance homomorphism counts case k=2, we spot some non-obvious regularity proper...

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If ∑ x∈N[v] f (x) ≥ 1 for each v ∈ V (G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f1, f2, . . . , fd} of signed dominating functions on Gwith the property that ∑d i=1 fi(x) ≤ 1 for each x ∈ V (G), is called a signed dominating fa...

PURPOSE Here we describe LifePrint, a sequence alignment-independent k-tuple distance method to estimate relatedness between complete genomes. METHODS We designed a representative sample of all possible DNA tuples of length 9 (9-tuples). The final sample comprises 1878 tuples (called the LifePrint set of 9-tuples; LPS9) that are distinct from each other by at least two internal and noncontigu...

Phylogenies are useful for organizing knowledge of biological diversity, for structuring classifications, and for providing knowledge of events that occurred during evolution. Different phylogenetic reconstruction techniques are available. In this paper Distance based technique is used. Distance measure is an important issue in phylogenetic analysis. Traditional approaches are time-consuming du...

In this paper, we provide a new upper bound for the α-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the α-rate domination number, which combines the concepts of...

Instead of studying evolutions governed by an evolutionary system starting at a given initial state on a prescribed future time interval, finite or infinite, we tackle the problem of looking both for a past interval [T − D, T ] of aperture (or length, duration) D and for the viable evolutions arriving at a prescribed terminal state at the end of the temporal window (and thus telescoping if more...

Phylogenies are useful for organizing knowledge of biological diversity, for structuring classifications, and for providing knowledge of events that occurred during evolution. Different phylogenetic reconstruction techniques are available. In this paper Distance based technique is used. Distance measure is an important issue in phylogenetic analysis. Traditional approaches are time-consuming du...

an equitable domination has interesting application in the contextof social networks. in a network, nodes with nearly equal capacitymay interact with each other in a better way. in the societypersons with nearly equal status, tend to be friendly. in thispaper, we introduce new variant of equitable domination of agraph. basic properties and some interesting results have beenobtained.