We study partially distance-regular graphs and partially walkregular graphs as generalizations of distance-regular graphs and walkregular graphs respectively. We conclude that the partially distanceregular graphs can be viewed as some extremal graphs of partially walk-regular graphs. In the special case that the graph is assumed to be regular with four distinct eigenvalues, a well known class o...

In this paper, we prove the following two theorems. Theorem 1 Let 0 denote a distance-regular graph with diameter d ≥ 3. Suppose E and F are primitive idempotents of 0, with cosine sequences σ0, σ1, . . . , σd and ρ0, ρ1, . . . , ρd , respectively. Then the following are equivalent. (i) The entry-wise product E ◦ F is a scalar multiple of a primitive idempotent of 0. (ii) There exists a real nu...

A partition λ of a positive integer n is a sequence λ1 λ2 λm 0 of integers such that ∑λi n. For a positive integer p, a partition λ λ1 λ2 λm (or its Young diagram) is called p-regular if it does not have p or more equal parts, i.e. if there does not exist t m p 1 with λt λt 1 λt p 1. Let F be a field of characteristic p 0. It is well known that irreducible representations of the symmetric group...

in this paper we construct antimagic labelings of the regular complete multipartite graphs and we also extend the construction to some families of regular graphs.

In this paper we study some additive properties of subsets of N : A subset A of N is called ksummable (k a positive integer) if A contains {n∈F xn |F ⊆ {0, . . . , k − 1}} for some k-term sequence of natural numbers x0 < x1 < x2 < · · · < xk−1. We say A ⊆ N is FS big if A is k-summable for each positive integer k. We say is A ⊆ N is infinite FS big if for each positive integer k, A contains {n∈...

Let G be a connected reductive linear algebraic group over C and let (; V) be a regular representation of G. There is a locally-nite representation (^ ; C V ]) on the aane algebra C V ] of V deened by ^ (g)f(v) = f(g ?1 v) for f 2 C V ]. Since G is reductive, (^ ; C V ]) decomposes as a direct sum of irreducible regular representations of G. The representation (; V) is said to be multiplicity f...

In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z)=1$. In this paper, we prove Becker's conjecture in the best-possible form; we show that the rational function $R(z)$ can be...

Consider a simple connected graph $$G = (V,E)$$ of order p and size q. For bijection $$f : E \to \{1,2,\ldots,q\}$$ , let $$f^+(u) \sum_{e\in E(u)} f(e)$$ where $$E(u)$$ is the set edges incident to u. We say f local antimagic labeling G if for any two adjacent vertices u v, we have \ne f^+(v)$$ . The minimum number distinct values $$f^+$$ taken over all denoted by $$\chi_{la}(G)$$ Let $$G[H]$$...

In this paper we shall determine, when 1 = 6, bounds for numbers f(k, I) and F{k, 1) defined as follows: f{k, l)/F(k, I) is defined to be the smallest integer n for which there exists a regular graph/Hamiltonian regular graph of valency k and girth I having n vertices. The problem of determining minimal regular graphs of given girth was first considered by Tutte [9]. Bounds for f(k, I) have bee...

by Let QU denote the class of anal>-tic functions which are regular and unbounded in ;zI < 1. According io D. GAIER and IV. MEYER-K~NIG [I] we call the radius Rq defined bJ+ x = r&c'!, 0 5 r <: 1 singular for f(x), if f(z) is unbounded in any sector /zI < 1, pl-E < arg 2 < < p7 + e Fith E > 0. h radius which is not singular for f(z) is called regular for f(z). In [l] it has been shown that if f...