PERULANGAN PADA DIGRAF HAMPIR MOORE

Digraf Moore adalah graf berarah (directed graph) atau digraf yang memiliki derajat d, diameter k, dan jumlah titik sebanyak n = 1 + d + d 2 + ... + d k atau disebut sebagai jumlah Moore. Telah diketahui di (Plesnik & Znam, 1974), dan (Bidges & Toueg, 1980) bahwa digraf Moore hanya ada pada kasus-kasus trivial yaitu untuk d = 1 (digraf lingkaran Ck+1) dan untuk k = 1 (digraf lengkap Kd+1). Penelitian baru-baru ini diarahkan pada menentukan keberadaan digraf seperti di atas dengan jumlah titik kurang satu dari jumlah Moore atau disebut digraf hampir Moore dan ditulis sebagai (d,k)-digraf. Digraf yang memiliki jumlah titik seperti di atas mengakibatkan munculnya konsep perulangan dan perulangan-diri. Penelitian Miller & Fris (1992) mendapatkan bahwa (d,2)-digraf selalu ada. Pertanyaan yang sangat penting untuk dijawab adalah berapa banyak (d,2)-digraf yang memiliki struktur berbeda untuk d tertentu? Penulis dan peneliti yang lain mengunakan konsep perulangan dan perulangan-diri seperti di Baskoro, etal (1995), Simanjuntak & Baskoro, (1999), Iswadi & Baskoro, (1999) dan Baskoro, etal (1998) untuk menjawab sebagian pertanyaan di atas. Penelitian kali ini akan mengali lebih dalam sifatsifat titik perulangan dan perulangan-diri dengan mengembangkan hasil yang telah didapat di Iswadi & Baskoro, (1999)

On The Existence of Non-Diregular Digraphs of Order Two Less than the Moore Bound

A communication network can be modelled as a graph or a directed graph, where each processing element is represented by a vertex and the connection between two processing elements is represented by an edge (or, in case of directed connections, by an arc). When designing a communication network, there are several criteria to be considered. For example, we can require an overall balance of the system. Given that all the processing elements have the same status, the flow of information and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element, that is, if there is a balance (or regularity) in the network. This means that the in-degree and out-degree of each vertex in a directed graph (digraph) must be regular. In this paper, we present the existence of digraphs which are not diregular (regular out-degree, but not regular in-degree) with the number of vertices two less than the unobtainable upper bound for most values of out-degree and diameter, the so-called Moore bound

Subdigraphs of Almost Moore Digraphs Induced by Fixpoints of an Automorphism

The degree/diameter problem for directed graphs is the problem of determining the largest possible order for a digraph with given maximum out-degree d and diameter k. An upper bound is given by the Moore bound M(d,k)=1+d+d^2+...+d^k$ and almost Moore digraphs are digraphs with maximum out-degree d, diameter k and order M(d,k)-1. In this paper we will look at the structure of subdigraphs of almost Moore digraphs, which are induced by the vertices fixed by some automorphism varphi. If the automorphism fixes at least three vertices, we prove that the induced subdigraph is either an almost Moore digraph or a diregular k-geodetic digraph of degree d'<d-1, order M(d',k)+1 and diameter k+1. As it is known that almost Moore digraphs have an automorphism r, these results can help us determine structural properties of almost Moore digraphs, such as how many vertices of each order there are with respect to r. We determine this for d=4 and d=5, where we prove that except in some special cases, all vertices will have the same order

Degree/diameter problem for mixed graphs

The Degree/diameter problem asks for the largest graphs given diameter and maximum degree. This problem has been extensively studied both for directed and undirected graphs, ando also for special classes of graphs. In this work we present the state of art of the degree/diameter problem for mixed graphs

Nonexistence of almost Moore digraphs of degrees 4 and 5 with self-repeats

An almost Moore (d,k)-digraph is a regular digraph of degree d>1, diameter k>1 and order N(d,k)=d+d2+⋯+dk. So far, their existence has only been shown for k=2, whilst it is known that there are no such digraphs for k=3, 4 and for d=2, 3 when k≥3. Furthermore, under certain assumptions, the nonexistence for the remaining cases has also been shown. In this paper, we prove that (4,k) and (5,k)-almost Moore digraphs with self-repeats do not exist for k≥5.Nacho López: Supported in part by grants PID2020-115442RB-I00 and 2021 SGR-00434. Arnau Messegué: Supported in part by grants Margarita Sala and 2021SGR-00434. Josep M. Miret: Supported in part by grants PID2021-124613OB-I00 and 2021 SGR-00434.Peer ReviewedPostprint (published version

Nonexistence of almost Moore digraphs of diameter four

Regular digraphs of degree d > 1, diameter k > 1 and order N(d; k) = d+ +dk will be called almost Moore (d; k)-digraphs. So far, the problem of their existence has only been solved when d = 2; 3 or k = 2; 3. In this paper we prove that almost Moore digraphs of diameter 4 do not exist for any degree dPostprint (published version

Multipartite Moore digraphs

We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that every vertex of a given partite set is adjacent to the same number $\delta$ of vertices in each of the other independent sets. We determine when a Moore multipartite digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a Moore $r$-partite digraph with interpartite outdegree $\delta>1$ and diameter $k=2m$ are obtained. In the case $\delta=1$, which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive