An independent set of vertices S of a graph dominates the graph efficiently if every vertex of the graph is either in S or has precisely one neighbour in S. In this paper we prove that a connected cubic vertex-transitive graph on a power of 2 vertices has a set that dominates it efficiently if and only if it is not isomorphic to a Möbius ladder. This is a preprint of an article accepted for pub...

Our paper has two goals: i) We propose the combinatorial approach to facilitate the calculation of the number of spanning trees for five new classes of graphs. ii) We use a new powerful operation (subdivision) to get larger graphs from a given graph. In particular, we derive the explicit formulas for the subdivision of ladder, fan, triangular snake, double triangular snake and the total graph o...

Ladder Wave Digital Filters is a class of IIR digital filters derived from double resistively terminated ladder networks, which gives them good stability and low sensitivity to variations in the element values. An approach to design maximally fast bit-serial ladder wave digital filters is presented based on the numerically equivalent state-space representation of the signal-flow graph. An algor...

A ladder lottery, known as “Amidakuji” in Japan, is a common way to choose one winner or to make an assignment randomly in Japan. Formally, a ladder lottery L of a permutation π = (x1, x2, . . . , xn) is a network with n vertical lines (lines for short) and many horizontal lines (bars for short) connecting two consecutive vertical lines. The top ends of lines correspond to π. See Fig. 1. Each n...

Let G = (V,E) be a nontrivial, finite, and connected graph. A function c from E to {1,2,...,k},k ∈ N, can considered as rainbow k-coloring if every two vertices x y in has an x- path. Therefore, no path's edges receive the same color; this condition is called “rainbow path”. The smallest positive integer k, designated by rc(G), connection number. Thus, k-coloring. Meanwhile, strong within for h...

An injective function f : V (G)→ {0, 1, 2, . . . , q} is an odd sum labeling if the induced edge labeling f∗ defined by f∗(uv) = f(u) + f(v), for all uv ∈ E(G), is bijective and f∗(E(G)) = {1, 3, 5, . . . , 2q − 1}. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper, we have studied the odd sum property of the subdivision of the triangular snake, quadrilatera...

The height of a trace is the height of the corresponding heap of pieces in Viennot's representation, or equivalently the number of factors in its Cartier-Foata decomposition. Let h(t) and jtj stand respectively for the height and the length of a trace t. We prove that the bivariate commutative series P t x h(t) y jtj is rational, and we give a nite representation of it. We use this result to ob...

We study Abelian sandpiles on graphs of the form G×I, where G is an arbitrary finite connected graph, and I ⊂ Z is a finite interval. We show that for any fixed G with at least two vertices, the stationary measures μI = μG×I have two extremal weak limit points as I ↑ Z. The extremal limits are the only ergodic measures of maximum entropy on the set of infinite recurrent configurations. We show ...