Codes from the Incidence Matrices of a zero-divisor Graphs

Afrika Matematika, 2018

We exhibit PD-sets for the binary and non-binary codes generated by incidence matrices of triangular graphs T n where n ≥ 5.

Algebra and Discrete Mathematics, 2021

Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We also present graph theoretic conditions for such linear codes to be Type I and Type II self-dual. Several examples of binary linear codes produced by well-known graph classes are given.

Indonesian Journal of Combinatorics

Let $B(G)$ be the incidence matrix of a graph $G$. The row in $B(G)$ corresponding to a vertex $v$, denoted by $s(v)$ is the string which belongs to $\Bbb{Z}_2^n$, a set of $n$-tuples over a field of order two. The Hamming distance between the strings $s(u)$ and $s(v)$ is the number of positions in which $s(u)$ and $s(v)$ differ. In this paper we obtain the Hamming distance between the strings generated by the incidence matrix of a graph. The sum of Hamming distances between all pairs of strings, called Hamming index of a graph is obtained.

European Journal of Combinatorics, 2007

The stabilizers of the minimum-weight codewords of dual binary codes obtained from the strongly regular graphs T (n) defined by the primitive rank-3 action of the alternating groups A n where n ≥ 5, on Ω {2} , the set of duads of Ω = {1, 2, . . . , n}, are examined.

2016

In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over Z_2 which has the consecutive 1's property (i.e., 1's are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over Z_2 which satisfy consecutive 1's property. The set of such vectors is called a coding sequence of...

2018

Hamming distance of a two bit strings u and v of length n is defined to be the number of positions of u and v with different digit. If G is a simple graph on n vertices and m edges and B is an edge–vertex incidence matrix of G, then every edge e of G can be labeled using a binary digit string of length n from the row of B which corresponds to the edge e. We discuss Hamming distance of two different edges of the graph G. Then, we present formulae for the sum of all Hamming distances between two different edges of G, particularly when G is a path, a cycle, and a wheel, and some composite graphs.

2018

A zero-divisor graph of a commutative ring R, denoted Γ(R), is a simple graph with vertex set being the set of non-zero zero-divisors of R and with (x, y) an edge if and only if xy = 0. In this paper we study about the zerodivisor graph Γ(R) ,where R is a finite commutative ring. Also we study the compressed zero-divisor graph Γc(R) of R AMS Subject Classification: 05C10, 05C12

2011

Mathematics and Statistics, 2022

The unity product graph of a ring R is a graph which is obtained by setting the set of unit elements of R as the vertex set. The two distinct vertices r i and r j are joined by an edge if and only if r i •r j = e. The subgraphs of a unity product graph which are obtained by the vertex and edge deletions are said to be its induced and spanning subgraphs, respectively. A subset C of the vertex set of induced (spanning) subgraph of a unity product graph is called perfect code if the closed neighbourhood of c, S 1 (c) forms a partition of the vertex set as c runs through C. In this paper, we determine the perfect codes in the induced and spanning subgraphs of the unity product graphs associated with some commutative rings R with identity. As a result, we characterize the rings R in such a way that the spanning subgraphs admit a perfect code of order cardinality of the vertex set. In addition, we establish some sharp lower and upper bounds for the order of C to be a perfect code admitted by the induced and spanning subgraphs of the unity product graphs.

European Journal of Combinatorics, 1999

A code in a graph is a non-empty subset C of the vertex set V of. Given C, the partition of V according to the distance of the vertices away from C is called the distance partition of C. A completely regular code is a code whose distance partition has a certain regularity property. A special class of completely regular codes are the completely transitive codes. These are completely regular codes such that the cells of the distance partition are orbits of some group of automorphisms of the graph. This paper looks at these codes in the Hamming Graphs and provides a structure theorem which shows that completely transitive codes are made up of either transitive or nearly complete, completely transitive codes. The results of this paper suggest that particular attention should be paid to those completely transitive codes of transitive type.