Revistas académicas de la Universidad Católica del Norte
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Chromatic coloring of distance graphs, III
No full textA graph G(Z, D) with vertex set Z is called an integer distance graph if its edge set is obtained by joining two elements of Z by an edge whenever their absolute difference is a member of D. When D = P or D ⊆ P where P is the set of all prime numbers then we call it a prime distance graph. After establishing the chromatic number of G(Z, P ) as four, Eggleton has classified the collection of graphs as belonging to class i if the chromatic number of G(Z, D) is i. The problem of characterizing the family of graphs belonging to class i when D is of any given size is open for the past few decades. As coloring a prime distance graph is equivalent to producing a prime distance labeling for vertices of G, we have succeeded in giving a prime distance labeling for certain class of all graphs considered here. We have proved that if D = {2, 3, 5, 7, 7th prime, 10th prime, 13th prime, 16th prime, (7 + j)th prime, ..., (4 + j)th prime for any s ∈ N}, then there exists a prime distance graph with distance set D in class 4 and if D = {2, 3, 5, 4th prime, 6th prime, 8th prime, (4 + j)th prime, ..., (2 + j)th prime for any s ∈ N} then there exists a prime distance graphs with distance set D in class 3. Further, we have also obtained some more interesting results that are either general or existential such as a) If D is a specific sequence of integers in arithmetic progression then there exist a prime distance graph with distance set D, b) If G is any prime distance graph in class i for 1 ≤ i ≤ 4 then G × K2 is also a prime distance graph in the respective class i, c) A countable union of disjoint copies of prime distance graph is again a prime distance graph, d) The Middle/Total graph of a path on n vertices is a prime distance graph. In addition we also provide a new different proof for establishing a fact that all cycles are prime distance graph
Boundedness and asymptotic Behaviour of Solutions of some second-order nonlinear stochastic differential equations with delay
No full textThis paper considers a certain second-order nonlinear stochastic differential equation with delay. Novel conditions for the existence of solutions that are uniformly bounded and ultimately bounded are obtained. Moreover, we also study the asymptotic behaviour of solutions for the considered equation. We employ Lyapunov’s second method via an appropriate complete Lyapunov functional to achieve these. Obtained results are new, and they improve and complement some existing relatively recent results in the literature. Finally, an example is provided to illustrate the obtained results
The automorphism groups of some token graphs
No full textThe token graphs of graphs have been studied at least from the 80’s with different names and by different authors. The Johnson graph J(n, k) is isomorphic to the k-token graph of the complete graph Kn. To our knowledge, the unique results about the automorphism groups of token graphs are for the case of the Johnson graphs. In this paper we begin the study of the automorphism groups of token graphs of another graphs. In particular we obtain the automorphism group of the k-token graph of the path graph Pn, for n 6≠ 2k. Also, we obtain the automorphism group of the 2-token graph of the following graphs: cycle, star, fan and wheel graphs
Some generalized Ostrowski type fractional integral inequalities for MT−convex functions with applications on special means
No full textSome generalized Ostrowski-type integral inequalities for r−times differentiable functions whose absolute values are MT−convex have been discussed. Moreover, some applications on special bivariate means are obtained
Monophonic graphoidal covering number of corona product graphs
No full textIn a graph G, a chordless path is called a monophonic path. A collection ψm of monophonic paths in G is called a monophonic graphoidal cover of G if every vertex of G is an internal vertex of at most one monophonic path in ψm and every edge of G is in exactly one monophonic path in ψm. The monophonic graphoidal covering number ηm(G) of G is the minimum cardinality of a monophonic graphoidal cover of G. In this paper, we find the monophonic graphoidal covering number of corona product of some standard graphs
Invariant bilinear forms under the operator group of order p³ with odd prime p
No full textFor an odd prime p, we formulate the number of all degree n representations of a group of order p3. And calculating the dimension of space of invariant bilinear forms corresponding to degree n representation over a field F which contains a primitive p3 root of unity. Here we also explicitly discussed the existence of a non-degenerate invariant bilinear form of the same space
Characterization and commuting probability of n-centralizer finite rings
No full textLet R be a finite ring. The commuting probability of R is the probability that any two randomly chosen elements of R commute. A ring R is called an n-centralizer ring if it has n distinct centralizers. In this paper, we characterize some n-centralizer finite rings and compute their commuting probabilities
A graph product and its applications in generating non-cospectral equienergetic graphs
No full textA new graph product is defined in this paper and several applications of this product are described. The adjacency matrix of the product graph is given and its complete spectrum in terms of the spectrum of constituent graphs are determined. Sequences of cospectral graphs can be generated from the known cospectral graphs using the new product. Several sequences of non-cospectral equienergtic graphs can also be generated as an application of the graph product defined
A note on general sum-connectivity index
No full textFor a simple finite graph G, general sum-connectivity index is defined for any real number α as χα(G) = , which generalises both the first Zagreb index and the ordinary sum-connectivity index. In this paper, we present some new bounds for the general sum-connectivity index. We also present relation between general sum-connectivity index and general Randić index
Ergodicity of commuting multioperators and holomorphic multioperators of multiplication
No full textIn this paper, the strong ergodic theorems are extended from the case of one bounded operator to the case of commuting multioperators. The authors show that in Grothendieck space with the Dunford-Pettis property, mean ergodic operator, and uniform ergodic operator are the same. We study when multioperators of multiplication on a weighted Banach space of holomorphic multi-functions are power bounded, mean ergodic, or uniformly mean ergodic