The first and second Zagreb indices of a graph are equal, respectively, to the sum of squares of the vertex degrees, and the sum of the products of the degrees of pairs of adjacent vertices. We now consider analogous graph invariants, based on the second degrees of vertices (number of their second neighbors), called leap Zagreb indices. A number of their basic properties is established.

In a graph G, the first and second degrees of a vertex v is equal to thenumber of their first and second neighbors and are denoted by d(v/G) andd 2 (v/G), respectively. The first, second and third leap Zagreb indices are thesum of squares of second degrees of vertices of G, the sum of products of second degrees of pairs of adjacent vertices in G and the sum of products of firs...

Routing is a problem domain with an infinite number of final-solutions. One of the possible approaches to solving such problems is using graph theory. This paper presents mathematical analysis methodologies based on circular graphs for solving a shortest path routing problem. The problem is focused on searching for the shortest path within a circular graph. Such a search coincides with the netw...

In this work we present three new recognition algorithms for circular-arc graphs and for two subclasses of this graph class. We give a linear-time recognition algorithm for circular-arc graphs based on the algorithm of Eschen and Spinrad [ES93, Esc97]. Our algorithm is simpler than the earlier linear-time recognition algorithm of McConnell [McC03], which is the only linear time recognition algo...

Suppose G is a graph, r is a positive real number and S(r) is a circle of perimeter r. For a positive real number t ≤ r, a (t, r)circular consecutive colour-list assignment L is a mapping that assigns to each vertex v of G an interval L(v) of S(r) of length t. A circular L-colouring of G is a mapping f : V (G) → S(r) such that for each vertex v, f(v) ∈ L(v) and for each edge uv, the distance be...

Cross-linking of proteins by advanced glycation endproducts (AGEs) causes a host of pathological conditions but their exact roles are unknown. Cross-linking lysyl AGEs were synthesized and incorporated into two types of collagen peptides. The utility of these cross-linked peptides for biochemical investigations was demonstrated by proteolysis studies and circular dichroism.

This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) of a graph and prove that they are equivalent. Then we prove that for any graph , . Examples are given to show that this bound is sharp in the sense that for any , there is a graph with . It is also proved that -degenerate graphs have . T...

Original scientific paper The Kalman filter is a very general method of filtering which can solve problems such as optimal estimation, prediction, noise filtering, and optimal control. A problem with detection of correct path of moving objects is the received noisy data. Therefore, it is possible that the information is incorrectly detected. There are Different methods to extract the correct da...