On Total Irregularity Strength of Double-Star and Related Graphs

AbstractLet G = (V, E) be a simple and undirected graph with a vertex set V and an edge set E. A totally irregular total k-labeling f : V ∪ E → {1, 2,. . ., k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and x1, their weights and are distinct, and for any two different edges xy and x1y1 their weights f (x) + f (xy) + f (y) and f (x1) + f (x1y1) + f (y1) are also distinct. A total irregularity strength of graph G, denoted byts(G), is defined as the minimum k for which G has a totally irregular total k-labeling. In this paper, we determine the exact value of the total irregularity strength for double-star S n,m, n, m ≥ 3 and graph related to it, that is a caterpillar S n,2,n, n ≥ 3. The results are and ts(S n,2,n) = n

Nilai Total Ketidakteraturan Titik Pada Amalgamasi Graf Prisma

It is not possible to determine the total vertex of irregular strength of all graphs. This study aims to ascertain the total vertex irregularity strength in prismatic graph amalgamation for n>=4. Determination of the total vertex irregularity strength in prismatic graph amalgamation is done by ascertaining the largest lower limit and the smallest upper limit. The lower limit is analyzed based on the graph properties and other supporting theorems, while the upper limit is analyzed by labeling the vertices and edges of the prismatic amalgamation graph. Based on the results of this study, the total vertex irregularity strength in prismatic graph amalgamation is obtained, namely (4(P2,n))=2n , for n>=4.Penentuan nilai total ketidakteraturan titik dari semua graf belum dapat dilakukan. Penelitian ini bertujuan untuk menentukan nilai total ketidakteraturan titik pada amalgamasi graf prisma untuk n>=4.  Penentuan nilai total ketidakteraturan titik pada amalgamasi graf prisma dilakukan dengan menentukan batas bawah terbesar dan batas atas terkecil. Batas bawah dianalisis berdasarkan sifat-sifat graf dan teorema pendukung lainnya, sedangkan batas atas dianalisis dengan pemberian label pada titik dan sisi pada amalgamasi graf prisma. Berdasarkan hasil penelitian ini diperoleh nilai total ketidakteraturan titik pada amalgamasi graf prisma, (4(P2,n))=2n, untuk n>=4

Nilai Total Ketidakteraturan-H pada Graf Cn x P3

AbstrakPenentuan nilai total ketidakteraturan dari semua graf belum dapat dilakukan secara lengkap. Penelitian ini bertujuan untuk menentukan nilai total ketidakteraturan-H pada graf Cn x P3 untuk n ≥ 3 yang isomorfik dengan . Penentuan nilai total ketidakteraturan-H pada graf Cn x P3 dengan menentukan batas bawah terbesar dan batas atas terkecil. Batas bawah dianalisis berdasarkan sifat-sifat graf dan teorema pendukung lainnya. Sedangkan batas atas dianalisa dengan pemberian label pada titik dan sisi pada graf Cn x P3.Berdasarkan hasil penelitian ini diperoleh nilai total ketidakteraturan-H pada graf ths(Cn x P3, C4)=.Kata kunci : Selimut-H, Nilai total ketidakteraturan-HAbstractThe determine of H-irregularity total strength in all graphs was not complete on graph classes. The research aims to determine alghorithm the H-irregularity total strength of graph Cn x P3 for n ≥ 3 with use H-covering, where H is isomorphic to C4. The determine of H-irregularity total strength of graph Cn x P3 was conducted by determining lower bound and smallest upper bound. The lower bound was analyzed based on graph characteristics and other supporting theorem, while the upper bound was analyzed by edge labeling and vertex labeling of graph Cn x P3.The result show that  the H-irregularity total strength of graph ths(Cn x P3, C4)=.Keyword : H-covering, H-irregularity total strengt

NILAI TOTAL KETIDAKTERATURAN-H PADA GRAF Cn x P3

Penentuan nilai total ketidakteraturan dari semua graf belum dapat dilakukan secara lengkap. Penelitian ini bertujuan untuk menentukan nilai total ketidakteraturan-H pada graf Cn x P3 untuk n ≥ 3. Penentuan nilai total ketidakteraturan-H pada graf Cn x P3 dengan menentukan batas bawah terbesar dan batas atas terkecil. Batas bawah dianalisis berdasarkan sifat-siat graf dan teorema pendukung lainnya. Sedangkan batas atas dianalisa dengan pemberian label pada titik dan sisi pada graf Cn x P3.Berdasarkan hasil penelitian ini diperoleh nilai total ketidakteraturan-H pada graf (Cn x P dengan n ≥ 3. Kata kunci : Selimut-H, Nilai total ketidakteraturan-

Conifold transitions via affine geometry and mirror symmetry

Mirror symmetry of Calabi-Yau manifolds can be understood via a Legendre duality between a pair of certain affine manifolds with singularities called tropical manifolds. In this article, we study conifold transitions from the point of view of Gross and Siebert. We introduce the notions of tropical nodal singularity, tropical conifolds, tropical resolutions and smoothings. We interpret known global obstructions to the complex smoothing and symplectic small resolution of compact nodal Calabi-Yaus in terms of certain tropical $2$-cycles containing the nodes in their associated tropical conifolds. We prove that the existence of such cycles implies the simultaneous vanishing of the obstruction to smoothing the original Calabi-Yau \emph{and} to resolving its mirror. We formulate a conjecture suggesting that the existence of these cycles should imply that the tropical conifold can be resolved and its mirror can be smoothed, thus showing that the mirror of the resolution is a smoothing. We partially prove the conjecture for certain configurations of nodes and for some interesting examples.Comment: 82 pages, 28 figures. Published version. The main conjecture (Conjecture 8.3) has been reformulated. We added Section 9.5 where we partially prove the conjecture in an example. Improved expositio

NILAI TOTAL KETIDAKTERATURAN-H PADA GRAF C4 x Pn

Nilai total ketidakteraturan-H pada graf C4 x Pn diperoleh dengan menentukan batas bawah terbesar dan batas atas terkecil. Batas bawah dianalisis berdasarkan sifat-sifat graf dan teorema pendukung lainnya. Sedangkan batas atas dianalisa dengan pemberian label pada titik dan sisi pada graf C4 x Pn . Misal G adalah suatu graf, selimut sisi dari G (edge covering) adalah koleksi subgraf H_1,H_2,...,H_t dari graf G sedemikian sehingga setiap sisi dari G termuat dalam paling sedikit satu subgraf H_(i ,)dimana i=1,2,…,t. Dalam hal ini, maka G disebut memiliki selimut (sisi) -〖(H〗_1,H_2,...,H_t) (〖(H〗_1,H_2,...,H_t)-(edgecovering). Jika setiap subgraf H_i isomorfik dengan suatu graf H, maka dikatakan G memiliki suatu selimut-H(H-covering). Jika setiap subgraf isomorfik dengan suatu graf, maka dikatakan memiliki suatu selimut –H (H-covering). Hasil penelitian ini diperoleh nilai total ketidakteraturan-H pada graf C4 x Pn ths(C_4×P_n)=⌈(4n+3)/8⌉ dengan n ≥ 3. Kata kunci : Selimut-H, Nilai total ketidakteratuan-

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

Modelling Orebody Structures: Block Merging Algorithms and Block Model Spatial Restructuring Strategies Given Mesh Surfaces of Geological Boundaries

This paper describes a framework for capturing geological structures in a 3D block model and improving its spatial fidelity given new mesh surfaces. Using surfaces that represent geological boundaries, the objectives are to identify areas where refinement is needed, increase spatial resolution to minimize surface approximation error, reduce redundancy to increase the compactness of the model and identify the geological domain on a block-by-block basis. These objectives are fulfilled by four system components which perform block-surface overlap detection, spatial structure decomposition, sub-blocks consolidation and block tagging, respectively. The main contributions are a coordinate-ascent merging algorithm and a flexible architecture for updating the spatial structure of a block model when given multiple surfaces, which emphasizes the ability to selectively retain or modify previously assigned block labels. The techniques employed include block-surface intersection analysis based on the separable axis theorem and ray-tracing for establishing the location of blocks relative to surfaces. To demonstrate the robustness and applicability of the proposed block merging strategy in a more narrow setting, it is used to reduce block fragmentation in an existing model where surfaces are not given and the minimum block size is fixed. To obtain further insight, a systematic comparison with octree subblocking subsequently illustrates the inherent constraints of dyadic hierarchical decomposition and the importance of inter-scale merging. The results show the proposed method produces merged blocks with less extreme aspect ratios and is highly amenable to parallel processing. The overall framework is applicable to orebody modelling given geological boundaries, and 3D segmentation more generally, where there is a need to delineate spatial regions using mesh surfaces within a block model.Comment: Keywords: Block merging algorithms, block model structure, spatial restructuring, mesh surfaces, subsurface modelling, geological structures, sub-blocking, boundary correction, domain identification, iterative refinement, geospatial information system. 27 page article, 26 figures, 6 tables, plus supplementary material (17 pages