On the randic index of graphs

© 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/For a given graph G = (V, E), the degree mean rate of an edge uv ¿ E is a half of the quotient between the geometric and arithmetic means of its end-vertex degrees d(u) and d(v). In this note, we derive tight bounds for the Randic index of G in terms of its maximum and minimum degree mean rates over its edges. As a consequence, we prove the known conjecture that the average distance is bounded above by the Randic index for graphs with order n large enough, when the minimum degree d is greater than (approximately) ¿1/3 , where ¿ is the maximum degree. As a by-product, this proves that almost all random (Erdos–Rényi) graphs satisfy the conjecturePeer ReviewedPostprint (author's final draft

Iterated line digraphs are asymptotically dense

© 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We show that the line digraph technique, when iterated, provides dense digraphs, that is, with asymptotically large order for a given diameter (or with small diameter for a given order). This is a well- known result for regular digraphs. In this note we prove that this is also true for non-regular digraphsPostprint (author's final draft

The degree/diameter problem in maximal planar bipartite graphs

The (¿;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ¿ and diameter D. We consider the (¿;D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (¿; 2) problem, the number of vertices is n = ¿+2; and for the (¿; 3) problem, n = 3¿¿1 if ¿ is odd and n = 3¿ ¿ 2 if ¿ is even. Then, we study the general case (¿;D) and obtain that an upper bound on n is approximately 3(2D + 1)(¿ ¿ 2)¿D=2¿ and another one is C(¿ ¿ 2)¿D=2¿ if ¿ D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (¿ ¿ 2)k if D = 2k, and 3(¿ ¿ 3)k if D = 2k + 1, for ¿ and D sufficiently large in both cases.Postprint (published version

The spectra of subKautz and cyclic Kautz digraphs

© 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Kautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz CK(d,l) and the subKautz sK(d,2) digraphs were recently introduced by Böhmová, Huemer and the author. In this paper we propose a new method to obtain the complete spectra of subKautz sK(d,2) and cyclic Kautz CK(d,3) digraphs, for all d=3, through the Hoffman–McAndrew polynomial and regular partitions. This approach can be useful to find the spectra of other families of digraphs with high regularity.Postprint (author's final draft

The degree/diameter problem in maximal planar bipartite graphs

The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.Peer ReviewedPostprint (published version

Sequence mixed graphs

A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and literated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed diagraph by identifying some vertices of certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.Postprint (author's final draft

Sufficient conditions for a digraph to admit a (1,=l)-identifying code

A (1, = `)-identifying code in a digraph D is a subset C of vertices of D such that all distinct subsets of vertices of cardinality at most ` have distinct closed in-neighbourhoods within C. In this paper, we give some sufficient conditions for a digraph of minimum in-degree d - = 1 to admit a (1, = `)- identifying code for ` ¿ {d -, d- + 1}. As a corollary, we obtain the result by Laihonen that states that a graph of minimum degree d = 2 and girth at least 7 admits a (1, = d)-identifying code. Moreover, we prove that every 1-in-regular digraph has a (1, = 2)-identifying code if and only if the girth of the digraph is at least 5. We also characterize all the 2-in-regular digraphs admitting a (1, = `)-identifying code for ` ¿ {2, 3}.Peer ReviewedPostprint (author's final draft

Identifying codes from the spectrum of a graph or digraph

Peer ReviewedPostprint (author's final draft

Inflammation: The Link between Neural and Vascular Impairment in the Diabetic Retina and Therapeutic Implications

Diabetic retinopathy; Inflammation; RetinaRetinopatia diabètica; Inflamació; RetinaRetinopatía diabética; Inflamación; RetinaThe etiology of diabetic retinopathy (DR) is complex, multifactorial and compromises all the elements of the retinal neurovascular unit (NVU). This diabetic complication has a chronic low-grade inflammatory component involving multiple inflammatory mediators and adhesion molecules. The diabetic milieu promotes reactive gliosis, pro-inflammatory cytokine production and leukocyte recruitment, which contribute to the disruption of the blood retinal barrier. The understanding and the continuous research of the mechanisms behind the strong inflammatory component of the disease allows the design of new therapeutic strategies to address this unmet medical need. In this context, the aim of this review article is to recapitulate the latest research on the role of inflammation in DR and to discuss the efficacy of currently administered anti-inflammatory treatments and those still under development.This study has been funded by Instituto de Salud Carlos III (ISCIII) through the projects ICI20/00129 and PI22/01670 and co-funded by the European Union. Hugo Ramos is the recipient of a grant from the Ministerio de Economía y Competitividad (BES-2017-081690)

What else can we do to prevent diabetic retinopathy?

Diabetic retinopathy; Modifiable risk factors; Neurovascular unitRetinopatía diabética; Factores de riesgo modificables; Unidad neurovascularRetinopatia diabètica; Factors de risc modificables; Unitat neurovascularThe classical modifiable factors associated with the onset and progression of diabetic retinopathy are the suboptimal control of blood glucose levels and hypertension, as well as dyslipidaemia. However, there are other less recognised modifiable factors that can play a relevant role, such as the presence of obesity or the abnormal distribution of adipose tissue, and others related to lifestyle such as the type of diet, vitamin intake, exercise, smoking and sunlight exposure. In this article we revisit the prevention of diabetic retinopathy based on modulating the modifiable risk factors, as well as commenting on the potential impact of glucose-lowering drugs on the condition. The emerging concept that neurodegeneration is an early event in the development of diabetic retinopathy points to neuroprotection as a potential therapeutic strategy to prevent the advanced stages of the disease. In this regard, the better phenotyping of very early stages of diabetic retinopathy and the opportunity of arresting its progression using treatments targeting the neurovascular unit (NVU) are discussed.Open Access Funding provided by Universitat Autonoma de Barcelona