On Diregularity of Digraphs of Defect at Most Two
نویسندگان
چکیده
Since Moore digraphs do not exist for k 6= 1 and d 6= 1, the problem of finding digraphs of out-degree d ≥ 2, diameter k ≥ 2 and order close to the Moore bound, becomes an interesting problem. To prove the non-existence of such digraphs or to assist in their construction (if they exist), we first may wish to establish some properties that such digraphs must possess. In this paper we consider the diregularity of such digraphs. It is easy to show that any digraph with out-degree at most d ≥ 2, diameter k ≥ 2 and order one or two less than Moore bound must have all vertices of out-degree d. However, establishing the regularity or otherwise of the in-degree of such a digraph is not easy. In this paper we prove that all digraphs of defect two are either diregular or almost diregular. Additionally, in the case of defect one we present a new, simpler and shorter, proof that a digraph of defect one must be diregular, and in the case of defect two and for d = 2 and k ≥ 3, we present an alternative proof that a digraph of defect two must be diregular. This research was partly supported by the Leverhulme Visiting Professorship of the second author.
منابع مشابه
On diregularity of digraphs of defect two
Since Moore digraphs do not exist for k 6= 1 and d 6= 1, the problem of finding the existence of digraph of out-degree d ≥ 2 and diameter k ≥ 2 and order close to the Moore bound becomes an interesting problem. To prove the non-existence of such digraphs, we first may wish to establish their diregularity. It is easy to show that any digraph with out-degree at most d ≥ 2, diameter k ≥ 2 and orde...
متن کاملMore skew-equienergetic digraphs
Two digraphs of same order are said to be skew-equienergetic if their skew energies are equal. One of the open problems proposed by Li and Lian was to construct non-cospectral skew-equienergetic digraphs on n vertices. Recently this problem was solved by Ramane et al. In this paper, we give some new methods to construct new skew-equienergetic digraphs.
متن کاملThe Italian domatic number of a digraph
An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...
متن کاملOn spectral radius of strongly connected digraphs
It is known that the directed cycle of order $n$ uniquely achieves the minimum spectral radius among all strongly connected digraphs of order $nge 3$. In this paper, among others, we determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $nge 4$.
متن کاملOn the Structure of Diregular Digraphs with Defect 1
The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k. It is known that digraphs of order M d;k do not exist for d > 1 and k > 1. In this paper we study digraphs of order M d;k ? 1, that is, digraphs with defect 1, denoted by (d; k)-digraphs. If G is a (d; k)-digraph, then for each vertex v of G there exists a vertex w (called the repeat of v) such th...
متن کامل