On K-trees and Special Classes of K-trees

The class of k-trees is defined recursively as follows: the smallest k-tree is the k-clique. If G is a graph obtained by attaching a vertex v to a k-clique in a k-tree, then G is also a k-tree. Trees, connected acyclic graphs, are k-trees for k = 1. We introduce a new parameter known as the shell of a k-tree, and from the shell special subclasses of k-trees, tree-like k-trees, are classified. Tree-like k-trees are generalizations of paths, maximal outerplanar graphs, and chordal planar graphs with toughness exceeding one. Let fs = fs( G) be the number of independent sets of cardinality s of G. Then the polynomial I(G; x) = [special characters omitted] fs(G)x s is called the independence polynomial. All rational roots of the independence polynomials of paths are found, and the exact paths whose independence polynomials have these roots are characterized. Additionally trees are characterized that have ?1/q as a root of their independence polynomials for 1 ? q ? 4. The well known vertex and edge reduction identities for independence polynomials are generalized, and the independence polynomials of k-trees are investigated. Additionally, sharp upper and lower bounds for fs of maximal outerplanar graphs, i.e. tree-like 2-trees, are shown along with characterizations of the unique maximal outerplanar graphs that obtain these bounds respectively. These results are extensions of the works of Wingard, Song et al., and Alameddine. Let M1 and M2 be the first and second Zagreb index respectively. Then the minimum and maximum M1 and M2 values for k-trees are determined, and the unique k-trees that obtain these minimum and maximum values respectively are characterized. Additionally, the Zagreb indices of tree-like k-trees are investigated

Degree-Based Topological Indices

The degree of a vertex of a molecular graph is the number of first neighbors of this vertex. A large number of molecular-graph-based structure descriptors (topological indices) have been conceived, depending on vertex degrees. We summarize their main properties, and provide a critical comparative study thereof. (doi: 10.5562/cca2294

The Tur\'an density of tight cycles in three-uniform hypergraphs

The Tur\'an density of an $r$-uniform hypergraph $\mathcal{H}$, denoted $\pi(\mathcal{H})$, is the limit of the maximum density of an $n$-vertex $r$-uniform hypergraph not containing a copy of $\mathcal{H}$, as $n \to \infty$. Denote by $\mathcal{C}_{\ell}$ the $3$-uniform tight cycle on $\ell$ vertices. Mubayi and R\"odl gave an ``iterated blow-up'' construction showing that the Tur\'an density of $\mathcal{C}_5$ is at least $2\sqrt{3} - 3 \approx 0.464$, and this bound is conjectured to be tight. Their construction also does not contain $\mathcal{C}_{\ell}$ for larger $\ell$ not divisible by $3$, which suggests that it might be the extremal construction for these hypergraphs as well. Here, we determine the Tur\'an density of $\mathcal{C}_{\ell}$ for all large $\ell$ not divisible by $3$, showing that indeed $\pi(\mathcal{C}_{\ell}) = 2\sqrt{3} - 3$. To our knowledge, this is the first example of a Tur\'an density being determined where the extremal construction is an iterated blow-up construction. A key component in our proof, which may be of independent interest, is a $3$-uniform analogue of the statement ``a graph is bipartite if and only if it does not contain an odd cycle''.Comment: 34 pages, 4 figures (final version accepted to IMRN plus a few comments in conclusion

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

Točkovni procesi u analizi zavisnih podataka

For a stationary, regularly varying and weakly dependent $\mathbb{R}$–valued time series $(X_i)_{i \in \mathbb{Z}}$, the language of point processes offers a nice probabilistic way to deduce the distribution of various functionals of the sample $X_1, \dots , X_n$ as $n \to \infty$. This includes functionals whose behavior for large $n$ essentially depends only on extreme $X_i$’s. The main idea is to establish a so–called complete convergence result, that is convergence of point processes $\Sigma_{i=1}^n \delta_{(i/n,X_i/a_n)}$ for a suitable sequence $(a_n)$ and then apply continuous mapping arguments. In this context, the limiting point process is a Poisson cluster process which can be fully described by the so–called tail process of $(X_i)$. However, in this type of point process convergence the information on the temporal order of extreme $X_i$’s belonging to the same cluster is lost in the limit due to scaling of time. As one of the main contributions of the thesis, we present a new type of complete convergence result which preserves this kind of information. It applies to stationary regularly varying time series satisfying standard extremal dependence assumptions. Our approach is based on dividing the sample $X_1, \dots , X_n$ into smaller blocks and then considering these blocks, instead of only individual $X_i$’s, as points of a point process on a certain infinite–dimensional Polish space. Along the way, we revisit the notion of vague convergence of measures relying on an abstract theory of bounded sets and discuss general Poisson approximation theory for point processes on Polish spaces. The order preserving convergence allows us to prove new limit results for record times and partial sums of the underlying time series. In particular, we show that rescaled record times, under an additional assumption, converge in distribution to a certain scale invariant compound Poisson process. This extends the well known result in the i.i.d. case. Furthermore, when the index of regular variation is in the interval (0, 2), we obtain a new functional limit theorem for partial sums which applies to a variety of time series for which standard convergence in the space of càdlàg functions cannot hold. The main novelty is that the convergence is placed in the larger space of so–called decorated càdlàg functions equipped with an extension of Skorohod’s $M_2$ topology. Corollaries of this result are discussed. Finally, we use the language of stationary regularly varying random fields to revisit the well known problem of local alignment of sequences. For that purpose, we extend the notion of the tail process and the corresponding point process convergence theory to $\mathbb{R}$–valued random fields indexed over $d$–dimensional integer lattice. In the course of this extension we introduce the concept of anchoring which further clarifies the link between the tail process and the asymptotic distribution of a cluster of extremes.Za stacionaran vremenski niz $(X_i)_{i \in \mathbb{Z}}$ s vrijednostima u $\mathbb{R}$ kaže se da je regularno varirajući ako su mu sve konačno–dimenzionalne distribucije višedimenzionalno regularno varirajuće. To svojstvo ekvivalentno je postojanju takozvanog repnog procesa pomoću kojeg se na intuitivan način mogu opisati ekstremalna svojstva tog vremenskog niza. Za takav niz, uz pretpostavku slabe zavisnosti, teorija točkovnih procesa nudi lijep vjerojatnosni pristup za određivanje distribucije raznih funkcionala uzorka $X_1, \dots , X_n$ kada $n \to \infty$. Tu spadaju funkcionali čije ponašanje za velike n u principu ovisi samo o ekstremnim vrijednostima niza $X_i$. Glavna ideja je prvo pokazati konvergenciju točkovnih procesa $\Sigma_{i=1}^n \delta_{(i/n,X_i/a_n)}$ za prikladan niz $(a_n)$ te na nju primijeniti takozvane tehnike neprekidnih preslikavanja. Budući da iz te iste točkovne konvergencije možemo odrediti ponašanje puno različitih funkcionala, takvu konvergenciju često zovemo potpuna točkovna konvergencija. Nadalje, u ovom kontekstu granični točkovni proces je Poissonov proces s klasterima koji je potpuno određen repnim procesom niza $X_i$. Ipak, u ovom obliku potpune točkovne konvergencije, zbog skaliranja vremena, u limesu se gubi informacija o vremenskom poretku ekstremnih observacija koje su dio istog klastera. Jedan od glavnih doprinosa ove teze je predstavljanje novog oblika potpune točkovne konvergencije koji čuva ovaj poredak. On vrijedi za stacionarne regularno varirajuće vremenske nizove uz standardne pretpostavke o ekstremalnoj zavisnosti. Naš pristup baziran je na dijeljenju uzorka $X_1, \dots , X_n$ na manje blokove i tretiranjem tih blokova, umjesto pojedinačnih observacija, kao točaka točkovnog procesa na određenom beskonačno–dimenzionalnom poljskom prostoru. Usput, dajemo alternativan pristup takozvanoj vague konvergenciji mjera koristeći apstraktnu teoriju ograničenih skupova i diskutiramo generalne uvjete pod kojima određeni točkovni procesi na poljskim prostorima konvergiraju po distribuciji prema Poissonovom procesu. Potpuna točkovna konvergencija koja čuva poredak omogućava nam da pokažemo nove granične rezultate za vremena rekorda i parcijalne sume vremenskog niza u pozadini. Točnije, pokazujemo da reskalirana vremena rekorda, uz dodatnu pretpostavku na repni proces niza, konvergiraju po distribuciji prema određenom složenom Poissonovom procesu koji je invarijantan na skaliranje. Ovo poopćuje poznati rezultat u slučaju niza nezavisnih i jednako distribuiranih slučajnih varijabli. Nadalje, kada je indeks regularne varijacije u intervalu (0, 2), pokazujemo novi funkcionalni granični teorem za parcijalne sume koji je primjenjiv na velik broj vremenskih nizova za koje standardna konvergencija u prostoru càdlàg funkcija ne može vrijediti. Glavna novina je da se konvergencija gleda u većem prostoru takozvanih dekoriranih càdlàg funkcija uz topologiju koja je ekstenzija Skorohodove $M_2$ topologije. Također, diskutiramo i korolare ovog rezultata. Na kraju, koristimo jezik stacionarnih regularno varirajućih slučajnih polja kako bi dali novi uvid u klasični problem lokalnog poravnanja dvaju nizova znakova. U tu svrhu, proširujemo pojam repnog procesa i teoriju potpune točkovne konvergencije na slučajna polja sa skupom indeksa $\mathbb{Z}^d$. U sklopu te ekstenzije predstavljamo ideju usidrenja koja dodatno razjašnjuje vezu između repnog procesa i granične distribucije klastera ekstremnih observacija