Inverse problem for Albertson irregularity index

Graph indices have attracted great interest as they give us numerical clues for several properties of molecules. Some indices give valuable information on the molecules under consideration using mathematical calculations only. For these reasons, the calculation and properties of graph indices have been in the center of research. Naturally, the values taken by a graph index is an important problem called the inverse problem. It requires knowledge about the existence of a graph having index equal to a given number. The inverse problem is studied here for Albertson irregularity index as a part of investigation on irregularity indices. A class of graphs is constructed to Show that the Albertson index takes all positive even integers. It has been proven that there exists at least one tree with Albertson index equal to every even positive integer but 4. The existence of a unicyclic graph with irregularity index equal to m is shown for every even positive integer m except 4. It is also shown that the Albertson index of a cyclic graph can attain any even positive integer.Publisher's Versio

Inverse problem for Albertson irregularity index

Graph indices have attracted great interest as they give us numerical clues for several properties of molecules. Some indices give valuable information on the molecules under consideration using mathematical calculations only. For these reasons, the calculation and properties of graph indices have been in the center of research. Naturally, the values taken by a graph index is an important problem called the inverse problem. It requires knowledge about the existence of a graph having index equal to a given number. The inverse problem is studied here for Albertson irregularity index as a part of investigation on irregularity indices. A class of graphs is constructed to Show that the Albertson index takes all positive even integers. It has been proven that there exists at least one tree with Albertson index equal to every even positive integer but 4. The existence of a unicyclic graph with irregularity index equal to m is shown for every even positive integer m except 4. It is also shown that the Albertson index of a cyclic graph can attain any even positive integer.Publisher's Versio

Archaeogenetic analysis of Neolithic sheep from Anatolia suggests a complex demographic history since domestication

Yurtman, ozer, Yuncu et al. provide an ancient DNA data set to demonstrate the impact of human activity on the demographic history of domestic sheep. The authors demonstrate that there may have been multiple domestication events with notable changes to the gene pool of European and Anatolian sheep since the Neolithic. Sheep were among the first domesticated animals, but their demographic history is little understood. Here we analyzed nuclear polymorphism and mitochondrial data (mtDNA) from ancient central and west Anatolian sheep dating from Epipaleolithic to late Neolithic, comparatively with modern-day breeds and central Asian Neolithic/Bronze Age sheep (OBI). Analyzing ancient nuclear data, we found that Anatolian Neolithic sheep (ANS) are genetically closest to present-day European breeds relative to Asian breeds, a conclusion supported by mtDNA haplogroup frequencies. In contrast, OBI showed higher genetic affinity to present-day Asian breeds. These results suggest that the east-west genetic structure observed in present-day breeds had already emerged by 6000 BCE, hinting at multiple sheep domestication episodes or early wild introgression in southwest Asia. Furthermore, we found that ANS are genetically distinct from all modern breeds. Our results suggest that European and Anatolian domestic sheep gene pools have been strongly remolded since the Neolithic

Zagreb indices of subdivision graphs

Bu çalışmada alt graflar tanıtılmış, r-alt graflar tanımlanmış ve bu alt grafların on çeşit Zagreb indeksleri hesaplanmış ve r-alt graflar için bazı eşitsizlikler verilmiştir. Bu uygulama Zagreb indekslerinin hesabında, grafların her bir köşesinin tek tek dereceleri ile uğraşmak yerine, sadece grafın kenar ve köşe sayılarının bilinmesinin yeterli olduğunu gösteren bir çalışmadır ve Zagreb indekslerinin hesabında büyük kolaylık sağlamaktadır. Bu tez dört bölümden oluşmaktadır. Birinci bölüm giriş bölümü olup bu bölümde konunun literatür özeti yapılmış ve çalışmanın ilerleyen bölümlerinde kullanılacak olan bazı temel kavramlar verilmiştir. İkinci bölümde birinci ve ikinci Zagreb indeksleri ile bunların eşindeksleri, birinci ve ikinci çarpımsal Zagreb indeksleri ile bunların eşindeksleri, total çarpımsal toplam Zagreb indeksi ile çarpımsal toplam Zagreb indeksi tanımlanarak bu indekslerin tümü için bazı sınırlar ve birbirleriyle ilişkilerini veren bazı eşitsizlikler verilmiştir. Üçüncü bölümde iyi bilinen yol graf, devir graf, yıldız graf, tam graf, iki parçalı tam graf ve tadpole grafların on çeşit Zagreb indeksleri hesaplanarak birbirleriyle ilişkilerini veren bazı sonuçlar elde edilmiştir. Dördüncü bölümde iyi bilinen bazı alt grafların ve r-alt grafların on çeşit Zagreb indeksleri hesaplanarak alt grafların çeşitli Zagreb indeksleri arasında birtakım eşitsizlikler verilmiştir. Son bölümde verilen tüm sonuçlar bu tez çalışmasında elde edilmiş orijinal sonuçlardır.In this work, subdivision graphs are recalled, r-subdivision graphs are defined and ten types of Zagreb indices of these graphs are calculated. This application shows that it is enough to know only the number of vertices and edges of the graphs, instead of dealing with the degrees of all vertices of the graphs and it provides great convenience for the calculation of the Zagreb indices. This thesis consists of four chapters. First chapter is introduction, and a brief summary of related literature and the necessary preliminaries are given in this chapter. Some basic concepts which will be used in the forthcoming chapters are introduced here. In the second chapter, Zagreb and multiplicative Zagreb indices and coindices of graphs, total multpilicative sum Zagreb index and multpilicative sum Zagreb index are introduced and some results and theorems for all these Zagreb indices are given. In the third chapter, ten types of Zagreb indices are calculated for some well-known graphs, such as path graph, cycle graph, star graph, complete graph, complete bipartite graph and tadpole graph and some results are obtained. In the fourth chapter, ten types of Zagreb indices of subdivision and r-subdivision graphs for some well-known graphs are given and some inequalities which shows the relations between several Zagreb indices of subdivision graphs are obtained

Zagreb indices and multiplicative Zagreb indices of double graphs of subdivision graphs

Let G be a simple graph. The subdivision graph and the double graph are the graphs obtained from a given graph G which have several properties related to the properties of G. In this paper, the first and second Zagreb and multiplicative Zagreb indices of double graphs, subdivision graphs, double graphs of the subdivision graphs and subdivision graphs of the double graphs of G are obtained. In particular, these numbers are calculated for the frequently used null, path, cycle, star, complete, complete bipartite or tadpole graph.Publisher's Versio

The effect of edge and vertex deletion on omega invariant

Recently the first and last authors defined a new graph characteristic called omega related to Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined in terms of a given degree sequence as a graph invariant and gives a lot of information on the realizability, number of realizations, connectedness, cyclicness, number of components, chords, loops, pendant edges, faces, bridges etc. of the family of realizations.In this paper, the effect of the deletion of vertices and edges from a graph on omega invariant is studied

Archaeogenetic analysis of Neolithic sheep from Anatolia suggests a complex demographic history since domestication

Sheep were among the first domesticated animals, but their demographic history is little understood. Here we analyzed nuclear polymorphism and mitochondrial data (mtDNA) from ancient central and west Anatolian sheep dating from Epipaleolithic to late Neolithic, comparatively with modern-day breeds and central Asian Neolithic/Bronze Age sheep (OBI). Analyzing ancient nuclear data, we found that Anatolian Neolithic sheep (ANS) are genetically closest to present-day European breeds relative to Asian breeds, a conclusion supported by mtDNA haplogroup frequencies. In contrast, OBI showed higher genetic affinity to present-day Asian breeds. These results suggest that the east-west genetic structure observed in present-day breeds had already emerged by 6000 BCE, hinting at multiple sheep domestication episodes or early wild introgression in southwest Asia. Furthermore, we found that ANS are genetically distinct from all modern breeds. Our results suggest that European and Anatolian domestic sheep gene pools have been strongly remolded since the Neolithic