Graphs with three and four distinct eigenvalues based on circulants

In this paper, we aim to address the open questions raised in various recent papers regarding characterization of circulant graphs with three or four distinct eigenvalues in their spectra. Our focus is on providing characterizations and constructing classes of graphs falling under this specific category. We present a characterization of circulant graphs with prime number order and unitary Cayley graphs with arbitrary order, both of which possess spectra displaying three or four distinct eigenvalues. Various constructions of circulant graphs with composite orders are provided whose spectra consist of four distinct eigenvalues. These constructions primarily utilize specific subgraphs of circulant graphs that already possess two or three eigenvalues in their spectra, employing graph operations like the tensor product, the union, and the complement. Finally, we characterize the iterated line graphs of unitary Cayley graphs whose spectra contain three or four distinct eigenvalues, and we show their non-circulant nature.Comment: 24 page

Maximal diameter of integral circulant graphs

Integral circulant graphs are proposed as models for quantum spin networks that permit a quantum phenomenon called perfect state transfer. Specifically, it is important to know how far information can potentially be transferred between nodes of the quantum networks modelled by integral circulant graphs and this task is related to calculating the maximal diameter of a graph. The integral circulant graph $ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, \ldots, n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d<n\}$. Motivated by the result on the upper bound of the diameter of $ICG_n(D)$ given in [N. Saxena, S. Severini, I. Shparlinski, \textit{Parameters of integral circulant graphs and periodic quantum dynamics}, International Journal of Quantum Information 5 (2007), 417--430], according to which $2|D|+1$ represents one such bound, in this paper we prove that the maximal value of the diameter of the integral circulant graph $ICG_n(D)$ of a given order $n$ with its prime factorization $p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, is equal to $r(n)$ or $r(n)+1$, where $r(n)=k + |\{ i \ | \alpha_i> 1,\ 1\leq i\leq k \}|$, depending on whether $n\not\in 4N+2$ or not, respectively. Furthermore, we show that, for a given order $n$, a divisor set $D$ with $|D|\leq k$ can always be found such that this bound is attained. Finally, we calculate the maximal diameter in the class of integral circulant graphs of a given order $n$ and cardinality of the divisor set $t\leq k$ and characterize all extremal graphs. We actually show that the maximal diameter can have the values $2t$, $2t+1$, $r(n)$ and $r(n)+1$ depending on the values of $t$ and $n$. This way we further improve the upper bound of Saxena, Severini and Shparlinski and we also characterize all graphs whose diameters are equal to $2|D|+1$, thus generalizing a result in that paper.Comment: 29 pages, 1 figur

The influence of residual fertilizing by phosphorus and potassium upon winter wheat grain yield and some soil chemical properties in crop rotation maize-soybean-winter wheat-sugar beat

U radu se iznose rezultati osmogodišnjih istraživanja gnojidbe na zalihu fosforom i kalijem za ozimu pšenicu u četveropoljnom mjembenom plodoredu kukuruz-soja-ozima pšenica-šećerna repa u dvije rotacije plodoreda na lesi-viranom tlu na pretaloženom lesu, lokalitet Lukač (Virovitica). Uz negnojenu varijantu i standardnu mineralnu gnojidbu, u pokusu su bile zastupljene dvije varijante gnojidbe na zalihu fosforom i kalijem za četverogodišnje razdoblje i jedna varijanta s gnojidbom na zalihu za dvogodišnje razdoblje. U odnosu na prinos zrna ozime pšenice učinkovitost gnojidbe na zalihu tijekom osmogodišnjeg istraživačkog razdoblja bila je promjenjiva. Prema prosječnim osmogodišnjim rezultatima pri gnojidbi na zalihu ipak su dobiveni viši prinosi u usporedbi sa standardnom mineralnom gnojidbom, pri čemu je važniju ulogu od kalija odigrao fosfor s obzirom na fiziološke potrebe pšenice. Isto tako, prema prosječnim osmogodišnjim vrijednostima, sadržaj biljci pristupačnog fosfora i kalija u tlu u svih je gnojenih varijanata, osim standardne mineralne gnojidbe, porastao signifikantno u odnosu na negnojenu varijantu. Glede zasićenosti bazama, nastale promjene prvenstveno su se očitovale njezinim smanjenjem, što, drugim riječima, znači da je mineralna gnojidba djelovala nepovoljno na sve one parametre koji su izravno povezani s reakcijom tla, kao i one koji neizravno iskazuju takva obilježja.The paper presents the results of eight-year investigations of residual fertilization with phosphorus and potassium for winter wheat grown in the four-year field crop rotation: maize-soybean-winter wheat-sugar beet in two rotations on luvic soil on redeposited loess, locality Lukač (Virovitica). Besides the unfertilized variant and standard mineral fertilization, the trial also included two treatments of residual fertilization with phosphorus and potassium for a four-year period and one residual fertilization treatment for a two-year period. Efficiency of residual fertilization with respect to the winter wheat grain yield varied over the eight-year research period. Nevertheless, according to average eight-year results, higher yields were obtained with residual fertilization compared to standard mineral fertilization, in which, due to wheat physiological requirements, phosphorus had a more important role than potassium. Also, according to average eight-year values, plant available phosphorus and potassium contents of the soil increased significantly in all fertilized variants, except for standard mineral fertilization, compared to the unfertilized control variant. In the case of base saturation, the induced changes were primarily reflected in its reduction; in other words, mineral fertilization had an adverse effect on all parameters directly related to soil reaction, as well as on the parameters indirectly expressing such characteristics

Neurological Manifestation of Fabry Disease – A Case Report

Fabry disease is an X-linked recessive glycolipid storage disease. It is caused by deficiency of the lysosomal enzyme a-galactosidase A and leads to the accumulation of the enzyme substrate, globotriasylceramide (Gb3) in many tissues including endothelial cells, pericytes and smooth muscle cells of blood vessels, renal epithelial cells, cardiac myocytes and numerous neuronal cells. In this report, we present 20-year-old male patient with ischemic stroke in pons. The case had previously been misdiagnosed as polimyositis and vasculitis. Angiokeratomas, neuropathic pain and ischemic stroke in young age suggested a Fabry disease. The diagnosis was confirmed biochemically and genetically. All young adults with stroke, especially if they have additional symptoms like angiokeratomas, proteinuria, neuropathic pain in toes and fingers should be tested for Fabry disease

The influence of residual fertilizing by phosphorus and potassium upon sugar beat root yield and some soil chemical properties in crop rotation maize-soybean-winter wheat-sugar beat

Na lesiviranom tlu na pretaloženom lesu na lokalitetu Lukač (Virovitica) provedena su višegodišnja istraživanja gnojidbe na zalihu fosforom i kalijem za šećernu repu u sklopu četverogodišnjeg mjembenog plodoreda u dvije rotacije. Uz negnojenu varijantu i standardnu mineralnu gnojidbu, istraživanja su obuhvatila gnojidbu na zalihu u dvije varijante za četverogodišnje razdoblje i jednu varijantu za dvogodišnje razdoblje, dakako s različitim izvorima fosfora, pa donekle i kalija. Što se tiče prinosa korijena, šećerna je repa vrlo povoljno reagirala na gnojidbu na zalihu fosforom i kalijem, pri čemu ne bi trebalo biti dvojbe da je u tome kalij odigrao ključnu ulogu. Izvjesna, ali ipak samo relativna prednost na strani je tripleksa i kalijeve soli pred kompleksnim gnojivom. Povećanje sadržaja biljci pristupačnog fosfora i kalija u tlu prema višegodišnjim prosječnim vrijednostima nije sukladno primijenjenim količinama, iako je značajnije u kalija. Ipak, u pojedinim godinama to je povećanje signifikantno pod utjecajem fosforno-kalijeve gnojidbe. U relativnom se smislu smanjuje uglavnom zasićenost adsorpcijskog kompleksa bazama u odnosu na kontrolu. Vrijednosti u pojedinim godinama vrlo su heterogene pod utjecajem fosforno-kalijeve gnojidbe.Residual fertilization with phosphorus and potassium was investigated on sugar beet grown in two rotations within the four-year field crop rotation. The several years long trial was set up on luvic soil on redeposited loess at the locality Lukač (Virovitica). Besides the unfertilized variant and standard mineral fertilization, the trial also included two variants of residual fertilization for a four-year period and one variant for a two-year period, naturally with different sources of phosphorus, and partly also potassium. As regards its root yields, sugar beet responded favourably to residual fertilization with phosphorus and potassium, in which potassium indisputably played the key role. It was found that triplex and potassium salt had an advantage, though only relative, over complex fertilizer. According to average several-years values, increase of plant available phosphorus and potassium content in the soil was not proportional to their quantities applied, though it was more significant for potassium. Nonetheless, this increase was significant in some years due to phosphorus-potassium fertilization. In a relative sense, mainly the base saturation of the cation exchange capacity was reduced compared to the control. Under the influence of phosphorus-potassium fertilization, very heterogeneous values were recorded in particular trial years

The Effect of Liming and Fertilization on Yields of Maize and Winter Wheat

Effect of different rates of hydrated lime and zeolite tuffs, as lime materials, mineral and organic fertilizers upon the yield of maize and winter wheat grain was studied in an exact field trial set up on Eutric Gleysol, near Karlovac, Central Croatia. The following crops were cultivated during the study period: 1999 and 2001 – maize, 1999/00 and 2001/02 – winter wheat. In the first investigation year, the highest yield of maize grain of 9.78 t ha-1 was achieved with the combination of the higher mineral fertilizer rate and the higher rate of farmyard manure. In the following year, the highest yield of winter wheat grain of 5.85 t ha-1 was achieved with the combination of the higher mineral fertilizer rate and the higher rate of hydrated lime. In the third and fourth investigation years, the highest yields of maize grain (10.05 t ha-1) and wheat (5.48 t ha-1) were recorded for the combination of the higher rates of mineral fertilizers and hydrated lime. The foregoing allows the conclusion that mineral and organic fertilization combined with hydrated lime is the optimal solution for increasing the yields of test crops

Basic Gravimetric Network of the Republic of Croatia

U radu se daje pregled dosadašnjih gravimetrijskih mjerenja na području Hrvatske te prikaz uspostave nove Osnovne gravimetrijske mreže (OGM) Republike Hrvatske. Ta se mreža sastoji od ukupno 42 točke, od čega je 6 apsolutnih gravimetrijskih točaka (Gravimetrijska mreža 0. reda) i 36 relativnih gravimetrijskih točaka (Gravimetrijska mreža I. reda). Nulta serija mjerenja u OGM-u obavljena je dvama relativnim gravimetrima Scintrex CG-3M i jednim Scintrex CG-5 u razdoblju od lipnja do kolovoza 2003. godine. U nastavku su prikazani obradba i izjednačenje tih mjerenja te je dana analiza rezultata za konačne vrijednosti ubrzanja sile teže temeljem provedenih izjednačenja. U svrhu dobivanja pouzdane transformacije između Potsdamskog sustava i gravimetrijskog sustava IGSN71 izračunani su transformacijski parametri te je dana njihova ocjena točnosti.In this paper a review of the past gravimetric measurements over the territory of Croatia and the description of the establishment of the Basic Gravimetric Network (BGN) of the Republic of Croatia are given. This network consists from totally 42 points; 6 absolute gravimetric points (0. Order Gravity Network) and 36 relative gravimetric points (I. Order Gravity Network). Zero series of measurements in BGN have been done using two relative gravimeters Scintrex CG-3M and one Scintrex CG-5 in the period between June and August 2003. In the continuation, the processing and the adjustment of those measurements, as well as the analysis of the results for the final gravity values are reported, based on the conducted adjustments. In order to get a reliable transformation between Potsdam and IGSN71 gravimetric datum, the transformation parameters have been calculated and their error estimates given