Large cycles in 4-connected graphs

Every 4-connected graph $G$ with minimum degree $\delta$ and connectivity $\kappa$ either contains a cycle of length at least $4\delta-\kappa-4$ or every longest cycle in $G$ is a dominating cycle.Comment: 4 page

On Dirac's Conjecture

Let $G$ be a 2-connected graph, $l$ be the length of a longest path in $G$ and $c$ be the circumference - the length of a longest cycle in $G$. In 1952, Dirac proved that $c>\sqrt{2l}$ and conjectured that $c\ge 2\sqrt{l}$. In this paper we present more general sharp bounds in terms of $l$ and the length $m$ of a vine on a longest path in $G$ including Dirac's conjecture as a corollary: if $c=m+y+2$ (generally, $c\ge m+y+2$) for some integer $y\ge 0$, then $c\ge\sqrt{4l+(y+1)^2}$ if $m$ is odd; and $c\ge\sqrt{4l+(y+1)^2-1}$ if $m$ is even.Comment: 6 pages, major revisio

Electron Wave Function in the Field of a One Dimensional Irregular Layered Structure

A method is proposed to find the wave function of an electron moving infinitely in the field of an arbitrary 1D layer structure with two different homogeneous semi-infinite boundaries. It is shown that in general the problem reduces to solution of a set of two linear difference equations. The proposed approach is discussed on a base of two cases: a structure of periodically placed identical rectangular potentials and a non-ordered structure with certain distortion of periodicity and potential identity.Comment: Submitted in Supperlattices and Microstructure

Evolutionary Hamiltonian Graph Theory

We present an alternative domain concerning mathematics to investigate universal evolution mechanisms by focusing on large cycles theory (LCT) - a simplified version of well-known hamiltonian graph theory. LCT joins together a number of $NP$-complete cycle problems in graph theory. $NP$-completeness is the kay factor insuring (by conjecture of Cook) the generation of endless developments and great diversity around large cycles problems. Originated about 60 years ago, the individuals (claims, propositions, lemmas, conjectures, theorems, and so on) in LCT continually evolve and adapt to their environment by an iterative process from primitive beginnings to best possible theorems based on inductive reasoning. LCT evolves much more rapidly than biosphere and has a few thousand pronounced species (theorems). Recall that life on earth with more than 2 million species was originated about 3.7 billion years ago and evolves extremely slowly. We show that all theorems in LCT have descended from some common primitive propositions such as "every complete graph is hamiltonian" or "every graph contains a cycle of length at least one" via improvements, modifications and three kinds of generalizations - closing, associating and extending. It is reasonable to review Darwinian mechanisms in light of LCT evolution mechanisms (especially inductive reasoning) including the origin and macroevolution disputable phenomena in the biosphere.Comment: 32 pages, improved versio

Long Cycles in 1-tough Graphs

In 1952, Dirac proved that every 2-connected graph with minimum degree $\delta$ either is hamiltonian or contains a cycle of length at least $2\delta$. In 1986, Bauer and Schmeichel enlarged the bound $2\delta$ to $2\delta+2$ under additional 1-tough condition - an alternative and more natural necessary condition for a graph to be hamiltonian. In fact, the bound $2\delta+2$ is sharp for a graph on $n$ vertices when $n\equiv 1(mod\ 3)$. In this paper we present the final version of this result which is sharp for each $n$: every 1-tough graph either is hamiltonian or contains a cycle of length at least $2\delta+2$ when $n\equiv 1(mod\ 3)$, at least $2\delta+3$ when $n\equiv 2(mod\ 3)$ or $n\equiv 1(mod\ 4)$, and at least $2\delta+4$ otherwise.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1204.651

Preliminary study of the missing mass spectra via the 12C(p,Ks0)^{12}C(p, K^0_s) and 12C(p,Λ)^{12}C(p,\Lambda) reactions at 10 GeV/c

The missing mass spectra for the $^{12}C(p, K^0_s)$ and $^{12}C(p,\Lambda)$ reactions have been studied by using of the propane bubble chamber(PBC) data from 700000 stereo photographs or $10^6$ inelastic interactions. The momentum spectrum of $\pi^-$ in range of 100-200 MeV/c have observed the significant enhancement from the p+C$\to \pi^-\Lambda$X(p+C$\to\pi^-K^0_s$X) reaction. The missing mass spectra have been observed signals for the $p(p,\pi^-K^0_s)$, $^3H(p,pK^0_s)$, p(p,$\Lambda)\pi^-\gamma$ and p(p,$\pi^-)\Lambda$ reactions. This experimental study will need to continue by using of a different identification methods for a reaction channels.Comment: 3 pages, 7 figures, Proc. EXA-2011,September 5th - 9th, 2011,Vienna, Austri

The spectral investigation of seven HII regions in Kazarian galaxies

According to SDSS DR5 spectra the spectrophotometric investigations of seven HII regions of six Kazarian galaxies are conducted. The abundances of heavy elements and helium and also quantity of ionizing stars and star formation rate are determined. The oxygen abundance 12+log(O/H) lies in the range 7.94 - 8.35. The mean log(S/O), log(Ar/O) and log(Ne/O) abundance ratios are equal to: -1.63, -2.37 and -0.78, respectively. The log(N/O) abundance ratio of investigated HII regions is in the interval -0.63 to -1.37. They occupy the same area in the diagram N/O - O/H as the high-excitation HII regions. Most likely, the ages of investigated HII regions are larger than 100-300 Myr, required for the enrichment in nitrogen by intermediate-mass stars. The star formation rate is one order as in HII regions in spiral and irregular galaxies, and is in the interval 0.05 - 0.81 M year-1.Comment: 10 pages, 4 figures, 4 tables, published in Astrophysics, Vol. 51, No. 1, 2008 (English translation of Astrofizika

My Research Visiting Card in Hamiltonian Graph Theory

We present eighteen exact analogs of six well-known fundamental Theorems (due to Dirac, Nash-Williams and Jung) in hamiltonian graph theory providing alternative compositions of graph invariants. In Theorems 1-3 we give three lower bounds for the length of a longest cycle $C$ of a graph $G$ in terms of minimum degree $\delta$, connectivity $\kappa$ and parameters $\bar{p}$, $\bar{c}$ - the lengths of a longest path and longest cycle in $G\backslash C$, respectively. These bounds have no analogs in the area involving $\bar{p}$ and $\bar{c}$ as parameters. In Theorems 11 and 12 we give two Dirac-type results for generalized cycles including a number of fundamental results (concerning Hamilton and dominating cycles) as special cases. Connectivity invariant $\kappa$ appears as a parameter in some fundamental results and in some their exact analogs (Theorems 3-10) in the following chronological order: 1972 (Chv\'{a}tal and Erd\"{o}s), 1981a (Nikoghosyan), 1981b (Nikoghosyan), 1985a (Nikoghosyan), 1985b (Nikoghosyan), 2000 (Nikoghosyan), 2005 (Lu, Liu, Tian), 2009 (Nikoghosyan), 2009a (Yamashita), 2009b (Yamashita), 2011a (Nikoghosyan), 2011b (Nikoghosyan).Comment: 11 page

On large kk-ended trees in connected graphs

A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let $\sigma_m$ be the minimum degree sum of an independent set of $m$ vertices. The main result (Theorem 2) provides a lower bound for $t_{k+1}$ in terms of $\sigma_m$ and relative orders: if $G$ is a connected graph and $k$, $\lambda$, $m$ are positive integers with $2\le m\le\min\{k,\lambda\}+1$ then either $t_{k+1} \ge \sigma_m +\lambda(k-m+1)+1$ or $t_k\ge t_{k+1}-\lambda+1$.Comment: 15 pages, major revision-

Exotic strange multibaryon states searches with Λ\Lambda- hyperon and Ks0K^0_s- meson systems in p+A collisions at momentum 10 GeV/c

Review for exotic strange multibaryon states were obseved in the effective mass spectra of: 1)$\Lambda \pi^+$, $\Lambda \pi^-$, $\Lambda p$, $\Lambda p p$, and $\Lambda K^0_S$, $K^0_S\pi^{\pm}$ and $K^0_Sp$ subsystems.The invariant mass of $\Lambda \pi^+$ and $K^0_S\pi^{\pm}$ spectra has observed well known $\Sigma^{*+}$(1385) and $K^{*\pm}$(892) resonances. The width of $\Sigma^{*-}(1385)$ for p+A reaction is two time larger than that presented in PDG. The cross section of $\Xi^-\to \Lambda \pi^-$ is 7-8 times larger than expected geometrical cross section in p+propane interaction. A few events detected on the photographs of the propane bubble chamber, were interpreted as S=-2 light and heavy $H^{0,+}$ dibaryons.Comment: 10 pages, 16 figures, Proc. Int. Conference Hadron Structure,2-7 September, Bratislava, 200