Let r and m be two integers such that r > m. Let H be a graph with order |H|, size e and maximum degree r such that 2e > |H|r−m. We find a best lower bound on spectral radius of graph H in terms of m and r. Let G be a connected r-regular graph of order |G| and k < r be an integer. Using the previous results, we find some best upper bounds (in terms of r and k) on the third largest eigenvalue th...

Let Γ be a triangle-free distance-regular graph with diameter d ≥ 3, valency k ≥ 3 and intersection number a2 6= 0. Assume Γ has an eigenvalue with multiplicity k. We show that Γ is 1-homogeneous in the sense of Nomura when d = 3 or when d ≥ 4 and a4 = 0. In the latter case we prove that Γ is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d = 5 the ...

A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial and algebraic) proofs of the fact that every edge-distance-regular graph Γ is distance-regular and homogeneous. More precisely, Γ is edge-distance-regular if and only if it is bipartite distance-reg...

A regular language is called union-free if it can be represented by a regular expression that does not contain the union operation. Every regular language can be decomposed into a union of a finite number of union-free languages (the so-called union-free decomposition). We call the number of components in the minimal union-free decomposition of a regular language the union width of the regular ...

We show that every regular language, via some DFA which accepts it, gives rise to a homomorphism from the semiring of polynomial functors to the semiring of n×n matrices over polynomial functors. Given some polynomial functor and a regular language, this homomorphism can be used to automatically derive a functor whose values have the same shape as those of the original functor, but whose sequen...

We introduce the notions of blocked, block-marked and block-deterministic regular expressions. We characterize block-deterministic regular expressions with determin-istic Glushkov block automata. The results can be viewed as a generalization of the characterization of one-unambiguous regular expressions with deterministic Glushkov automata. In addition, when a language L has a block-determinist...

we show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. as aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. this generalizes and simplifies a result of  dung and   smith. as another consequen...

let $n,t_1,...,t_k$ be distinct positive integers. a toeplitz graph $g=(v, e)$ denoted by $t_n$ is a graph, where $v ={1,...,n}$ and $e= {(i,j) : |i-j| in {t_1,...,t_k}}$.in this paper, we present some results on decomposition of toeplitz graphs.

the echinoid fauna of basab area (aptian), northwest of kerman, is fairly rich and diverse, represented by pygaulidae, acropeltidae, emiratiidae, toxasteridae, holectypidae, stomechinidae and saleniidae. four regular echinoid species, among which three are new, assigned to genus tetragramma agassiz, are described and illustrated in the present paper. the new species are tetragramma basabensis, ...