On H-irregularity Strengths of G-amalgamation of Graphs

A simple graph G=(V(G),E(G)) admits an H-covering if every edge in E(G) belongs at least to one subgraph of G isomorphic to a given graph H. Then the graph G admitting H-covering admits an H-irregular total k-labeling f: V(G) U E(G) \to {1, 2, ..., k} if for every two different subgraphs H\u27 and H\u27\u27 isomorphic to H there is $wt_{f}(H\u27) \neq wt_{f}(H\u27\u27)$, where $wt_{f}(H)= \sum \limits_{v\in V(H)} f(v) + \sum \limits_{e \in E(H)} f(e)$ is the associated H-weight. The minimum k for which the graph G has an H-irregular total k-labeling is called the total H-irregularity strength of the graph G.In this paper, we obtain the precise value of the total H-irregularity strength of G-amalgamation of graphs

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

Computing the Edge Irregularity Strengths of Chain Graphs and the Join of Two Graphs

In computer science, graphs are used in variety of applications directly or indirectly. Especially quantitative labeled graphs have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V,E) with the vertex set V and the edge set E, a vertex k-labeling $\phi: V \rightarrow \{1,2,\dots, k\}$ is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f their $w_\phi(e) \ne w_\phi(f)$, where the weight of an edge $e=xy \in E(G)$ is $w_\phi(xy)=\phi(x)+\phi(y)$. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine the edge irregularity strengths of some chain graphs and the join of two graphs. We introduce a conjecture and open problems for researchers for further research

Wall-Crossing in Coupled 2d-4d Systems

We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N=2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of class S, that is, for those theories obtained by compactifying the six-dimensional (0,2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A_1 theories of class S. Finally, we indicate how our results can be used to produce solutions to the A_1 Hitchin equations on the Riemann surface C.Comment: 170 pages, 45 figure

Fermion condensation and super pivotal categories

We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call "m-type" and "q-type" particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if $\mathcal{C}$ is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies $\textbf{Tube}(\mathcal{C}/\psi) \cong \mathcal{C} \times (\mathcal{C}/\psi)$. We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the $SO(3)_6$ theory, and the $\frac{1}{2}\text{E}_6$ theory, and compute the quasiparticle excitation spectrum in each of these examples.Comment: 161 pages; v2: corrected typos (including 18 instances of "the the") and added some reference

Toda Systems, Cluster Characters, and Spectral Networks

We show that the Hamiltonians of the open relativistic Toda system are elements of the generic basis of a cluster algebra, and in particular are cluster characters of nonrigid representations of a quiver with potential. Using cluster coordinates defined via spectral networks, we identify the phase space of this system with the wild character variety related to the periodic nonrelativistic Toda system by the wild nonabelian Hodge correspondence. We show that this identification takes the relativistic Toda Hamiltonians to traces of holonomies around a simple closed curve. In particular, this provides nontrivial examples of cluster coordinates on $SL_n$-character varieties for $n > 2$ where canonical functions associated to simple closed curves can be computed in terms of quivers with potential, extending known results in the $SL_2$ case.Comment: 37 pages; Minor updates from previous versio

Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli Spaces

We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. On the other hand Gross and Siebert have succeeded in determining instanton corrections of complex structures of Calabi-Yau varieties in the context of mirror symmetry from a singular affine structure with additional data. We will show that the two instanton correction problems are equivalent in an appropriate sense via the identification of the wall crossing formulas in the metric problem with consistency conditions in the complex structure problem. This result provides examples of Calabi-Yau varieties where the instanton correction (in the sense of mirror symmetry) of metrics and complex structures can be determined.Comment: 160 pages. Revised version. References and acknowledgement added. Minor mistakes and typos corrected. Exposition improve

Cluster varieties from Legendrian knots

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.Comment: 47 page