A Group Labeled Graph is a pair (G,Λ) where G is an oriented graph and Λ is a mapping from the arcs of G to elements of a group. A (not necessarily directed) cycle C is called non-null if for any cyclic ordering of the arcs in C, the group element obtained by ‘adding’ the labels on forward arcs and ‘subtracting’ the labels on reverse arcs is not the identity element of the group. Non-null cycle...

In this paper, we determine the Randić energy of m-splitting graph, m-shadow graph and m-duplicate a given m being an arbitrary integer. Our results allow construction infinite sequence graphs having same energy. Further, some invariants like degree Kirchhoff index, Kemeny’s constant number spanning trees special graphs. From our results, indicate how to obtain infinitely many pairs equienerget...

We define cooperative games on general graphs and generalize Lloyd S. Shapley's celebrated allocation formula for those in terms of stochastic path integral driven by the associated Markov chain each graph. then show that value operator, one player defined integral, coincides with player's component game which is solution to least squares (or Poisson's) equation, light combinatorial Hodge decom...

Let G = (V,E) be a simple graph with the vertex set V and the edge set E. G is a sum graph if there exists a labelling f of the vertices of G into distinct positive integers such that uv ∈ E if and only if f(w) = f(u) + f(v) for some vertex w ∈ V . Such a labelling f is called a sum labelling of G. The sum number σ(G) of G is the smallest number of isolated vertices which result in a sum graph ...

OF THE DISSERTATION Explicit Bases of Motives over Number Fields with Application to Feynman Integrals by Yu, Yang Doctor of Philosophy in Mathematics, Washington University in St. Louis, 2016. Professor Matthew Kerr, Chair Let K∗(k) be the algebraic K-theory of a number field k and MT (k) the Tannakian category of mixed Tate motives over k. Then ExtMT (k)(Q(0),Q(n)) = K2n−1(k) ⊗ Q. Periods of ...

The integral circulant graph Xn(D) has the vertex set Zn = {0, 1, 2, . . . , n − 1} and vertices a and b are adjacent, if and only if gcd(a − b, n) ∈ D, where D = {d1, d2, . . . , dk} is a set of divisors of n. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper, we deal wi...

In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we obtain an existence theorem for a special type of fractional integral equation. Moreover, we establish a result on the convergence of successive approximation...

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system τ , arising from the universal family of Calabi-Yau hypersurfaces Ya in a G-variety X of dimension n. First, we construct a natural topological correspondence between relative cycles in Hn(X − Ya,∪D − Ya) bounded by the union of G-invariant divisors ∪D in X to t...

The transient field due to canonical sources in planarly layered media is obtained using a leaky-wave analysis. By proper choice of integration paths in both the complex frequency and complex wavenumber planes, transient fields are obtained exactly as a temporal inversion integral over a discrete sum of residues. The residues include both proper and improper surface-wave modes, analytically con...