Suppose G is a connected, k-regular graph such that Spec…G† ˆ Spec…G† where G is a distance-regular graph of diameter d with parameters a1 ˆ a2 ˆ ˆ adÿ1 ˆ 0 and ad > 0; i.e., a generalized odd graph, we show that G must be distance-regular with the same intersection array as that of G in terms of the notion of Ho ̈man Polynomials. Furthermore, G is isomorphic to G if G is one of the odd polygon ...

The edge domination number γe(G) of a graph G is the minimum size maximal matching in G. It well known that this parameter computationally very hard, and several approximation algorithms heuristics have been studied. In present paper, we provide best possible upper bounds on for regular non-regular graphs terms their order maximum degree. Furthermore, discuss algorithmic consequences our result...

In this note we examine the connection between vertices of high eccentricity and the existence of k-factors in regular graphs. This leads to new results in the case that the radius of the graph is small (≤ 3), namely that a d-regular graph G has all k-factors, for k|V (G)| even and k ≤ d, if it has at most 2d+2 vertices of eccentricity > 3. In particular, each regular graph G of diameter ≤ 3 ha...

In this paper we consider the algebra (A, f, g), where f and g are m-ary and n-ary operations on the set A. We define the concept of regular algebra and we prove that for a regular medial algebra (A, f, g) there exists a commutative semigroup (A,+) such that the operations f and g have linear representations on (A,+). As a corollary, we show that for a regular medial algebra (A,F ) there exists...

We consider the dihamiltonian decomposition problem for 3-regular graphs. A graph G is dihamiltonian decomposable if in the digraph obtained from G by replacing each edge of G as two directed edges, the set of edges are partitioned into 3 edge-disjoint directed hamiltonian cycles. We suggest some conditions for dihamiltonian decomposition of 3-regular graphs: for a 3-regular graph G, it is diha...

with the left regular representation of G on L(G) corresponding to the action on the factor Vi and the right regular representation corresponding to the dual action on the factor V ∗ i . Under the left regular representation L (G) decomposes into irreducibles as a sum over all irreducibles, with each one occuring with multiplicity dim Vi (which is the dimension of V ∗ i ). To make things simple...

Let \((X,\mathscr {B}, \mu ,T,d)\) be a measure-preserving dynamical system with exponentially mixing property, and let \(\mu \) an Ahlfors s-regular probability measure. The covering problem concerns the set E(x) of points which are covered by orbits \(x\in X\) infinitely many times. We prove that Hausdorff dimension intersection any regular fractal G \(\dim _\mathrm{H}G>s-\alpha equals _\math...

Let G be a given graph (modelling a communication network) which we assume suuers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 ? p). In particular we are interested in ro-bustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our r...

11. Assume that (ρ, V ) is an irreducible regular representation of the linear algebraic group G. Fix v∗ ∈ V ∗ with v∗ 6= 0. For v ∈ V let φv ∈ Aff(G) be the representative function φv(g) = 〈v∗, ρ(g)v〉. Let E = {φv : v ∈ V } and let T : V → E be the map Tv = φv. Prove that T is a bijective linear map and that Tρ(g) = R(g)T for all g ∈ G, where R(g)f(x) = f(xg) for f ∈ Aff(G). Thus every irreduc...

It is known that, if G is a triply transitive permutation group on a finite set X with a regular normal subgroup N, then |N| = 2 (d ̂ 2) or |N| = 3. (See [12; Theorem 11.3].) If N is a regular normal subgroup of a permutation group G on X, xeX, and Gx is the stabiliser of x, then Gx £ G/JV (as abstract group), and so G has a representation on a set Y such that, for xeX, the representations of Gx...