Some notes on the threshold graphs

AbstractIn this paper we consider threshold graphs (also called nested split graphs) and investigate some invariants of these graphs which can be of interest in bounding the largest eigenvalue of some graph spectra

On eigenvalue inequalities of a matrix whose graph is bipartite

Abstract We consider the set of real zero diagonal symmetric matrices whose underlying graph, if not told otherwise, is bipartite. Then we establish relations between the eigenvalues of such matrices and those arising from their bipartite complement. Some accounts on interval matrices are provided. We also provide a partial answer to the still open problem posed in (Zhan in SIAM J. Matrix Anal. Appl. 27:851–860, 2006)

Spectral characterization of families of split graphs

An upper bound for the sum of the squares of the entries of the principal eigenvector corresponding to a vertex subset inducing a k-regular subgraph is introduced and applied to the determination of an upper bound on the order of such induced subgraphs. Furthermore, for some connected graphs we establish a lower bound for the sum of squares of the entries of the principal eigenvector corresponding to the vertices of an independent set. Moreover, a spectral characterization of families of split graphs, involving its index and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. In particular, the complete split graph case is highlighted

Relations between (κ, τ)-regular sets and star complements

Let G be a finite graph with an eigenvalue μ of multiplicity m. A set X of m vertices in G is called a star set for μ in G if μ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (k ,t)-regular if it induces a k -regular subgraph and every vertex not in the subset has t neighbors in it. We investigate the graphs having a (k,t)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples

GRAMON Database: Ten Years of Beryllium-7 Specific Activity Measurements

U radu je predstavljena novoformirana baza podataka GRAMON (Ground Air Radioactivity Monitoring) koja sadrži rezultate merenja radioaktivnosti u vazduhu na sedam lokacija: Beograd (Srbija), Ljubljana i Krško (Slovenija), Sarajevo (Bosna i Hercegovina), Podgorica (Crna Gora), Skoplje i Bitola (Severna Makedonija). Iz baze su za sve lokacije preuzete mesečne vrednosti specifične aktivnosti prirodnog radionuklida berilijuma-7 i potom analizirani deskriptivni statistički parametri od januara 2010. do decembra 2019. godine. Srednje vrednosti specifične aktivnosti berilijuma-7 tokom ovog perioda kreću se od 3,32 mBq/m3 u Sarajevu do 5,93 mBq/m3 u Beogradu. Koeficijent varijacije najmanji je za Krško (37,7%) i Sarajevo (38,6%), a najveći za Beograd (54,8%) i Bitolu (72,4%). Ipak, 3σ-opsezi svih sedam vremenskih serija se preklapaju. U daljoj statističkoj analizi biće ispitano da li među ovim vremenskim serijama postoje značajne razlike.This paper presents a recently established Ground Air Radioactivity Monitoring (GRAMON) database that contains the results of radioactivity measurements in the air at seven locations: Belgrade (Serbia), Ljubljana and Krško (Slovenia), Sarajevo (Bosnia and Herzegovina), Podgorica (Montenegro), Skopje and Bitola (North Macedonia). Monthly values of specific activity of the natural radionuclide beryllium-7 were selected from the database and descriptive statistical parameters were analyzed for each location from January 2010 to December 2019. The mean values of the specific activity of beryllium-7 over this period range from 3.32 mBq/m3 in Sarajevo to 5.93 mBq/m3 in Belgrade. The coefficient of variation is the least for Krško (37.7%) and Sarajevo (38.6%), and the largest for Belgrade (54.8%) and Bitola (72.4%). Still, the 3σ-intervals of all seven time series overlap. Further statistical analysis will investigate whether there are any significant differences among these time series.XXXII Simpozijum Društva za zaštitu od zračenja Srbije i Crne Gore, 4-6. oktobar 2023; Budva, Crna GoraProceedings: [https://vinar.vin.bg.ac.rs/handle/123456789/11602

Introducing a regional database of radioactivity in the air – GRAMON

Ground Air Radioactivity Monitoring (GRAMON) database is a recently established collection containing activity concentrations of gamma emitters in aerosol samples. The measurements come from Serbia (sampling site Belgrade), Slovenia (sampling sites Ljubljana and Krško), Bosnia and Herzegovina (sampling site Sarajevo), Montenegro (sampling site Podgorica), and North Macedonia (sampling sites Skopje and Bitola), thus covering the northern and central parts of the Balkan Peninsula. As a database arising from the monitoring programmes in several countries, GRAMON is not fully homogeneous in terms of the radionuclides and time periods studied. For example, the beryllium-7 records are available for all sampling sites, while the lead-210 records only in Serbia, Slovenia, and Bosnia and Herzegovina. The time series for Serbia and Slovenia began in 1991, for Montenegro and North Macedonia in 2008, and for Bosnia and Herzegovina in 2010. However, sampling, sample preparation, and measurement procedures across the sites and laboratories are similar. In brief, aerosol samples are collected on filter papers using air samplers. Activity concentrations of radionuclides are determined by standard gamma spectrometry using high-purity germanium detectors. The time series contain monthly mean activity concentrations. Since only some of the GRAMON records have been previously published, this database provides a source for radioactivity research in the region that has been underrepresented in large-scale studies. We further hope to expand the number of contributing laboratories and cover a wider region of Europe, especially its southern and eastern parts.Book of abstract

Inequalities for Laplacian Eigenvalues of Signed Graphs with Given Frustration Number

Balanced signed graphs appear in the context of social groups with symmetric relations between individuals where a positive edge represents friendship and a negative edge represents enmities between the individuals. The frustration number f of a signed graph is the size of the minimal set F of vertices whose removal results in a balanced signed graph; hence, a connected signed graph G˙ is balanced if and only if f=0. In this paper, we consider the balance of G˙ via the relationships between the frustration number and eigenvalues of the symmetric Laplacian matrix associated with G˙. It is known that a signed graph is balanced if and only if its least Laplacian eigenvalue μn is zero. We consider the inequalities that involve certain Laplacian eigenvalues, the frustration number f and some related invariants such as the cut size of F and its average vertex degree. In particular, we consider the interplay between μn and f

Efficient Algorithm for Generating Maximal L-Reflexive Trees

The line graph of a graph G is another graph of which the vertex set corresponds to the edge set of G, and two vertices of the line graph of G are adjacent if the corresponding edges in G share a common vertex. A graph is reflexive if the second-largest eigenvalue of its adjacency matrix is no greater than 2. Reflexive graphs give combinatorial ground to generate two classes of algebraic numbers, Salem and Pisot numbers. The difficult question of identifying those graphs whose line graphs are reflexive (called L-reflexive graphs) is naturally attacked by first answering this question for trees. Even then, however, an elegant full characterization of reflexive line graphs of trees has proved to be quite formidable. In this paper, we present an efficient algorithm for the exhaustive generation of maximal L-reflexive trees