Hyperbolicity on Graph Operators

A graph operator is a mapping F : Gamma → Gamma 0 , where Gamma and Gamma 0 are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Lambda(G); subdivision graph, S(G); total graph, T(G); and the operators R(G) and Q(G). In particular, we get relationships such as delta(G) ≤ delta(R(G)) ≤ delta(G) + 1/2, delta(Lambda(G)) ≤ delta(Q(G)) ≤ delta(Lambda(G)) + 1/2, delta(S(G)) ≤ 2delta(R(G)) ≤ delta(S(G)) + 1 and delta(R(G)) − 1/2 ≤ delta(Lambda(G)) ≤ 5delta(R(G)) + 5/2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.Supported in part by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain

Supersymmetry on Graphs and Networks

We show that graphs, networks and other related discrete model systems carry a natural supersymmetric structure, which, apart from its conceptual importance as to possible physical applications, allows to derive a series of spectral properties for a class of graph operators which typically encode relevant graph characteristics.Comment: 11 pages, Latex, no figures, remark 4.1 added, slight alterations in lemma 5.3, a more detailed discussion at beginning of sect.6 (zero eigenspace

Smarandache-Zagreb Index on Three Graph Operators

Many researchers have studied several operators on a connected graph in which one make an attempt on subdivision of its edges. In this paper, we show how the Zagreb indices, a particular case of Smarandache-Zagreb index of a graph changes with these operators and extended these results to obtain a relation connecting the Zagreb index on operators

On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio