On identifying codes in partial linear spaces

Cornell University - arXiv, 2018

Let G be a simple graph without isolated vertices. For a vertex i in G, the degree d i is the number of vertices adjacent to i and the average 2-degree m i is the mean of the degrees of the vertices which are adjacent to i. The sequence of pairs (d i , m i) is called the sequence of degree pairs of G. We provide some necessary conditions for a sequence of real pairs (a i , b i) of length n to be the degree pairs of a graph of order n. A graph G is called pseudo k-regular if m i = k for every vertex i while d i is not a constant. Let N (k) denote the minimum number of vertices in a pseudo k-regular graph. We utilize the above necessary conditions to find all pseudo 3-regular graphs of orders no more than 10, and all pseudo k-regular graphs of order N (k) for k up to 7. We give bounds of N (k) and show that N (k) is at most k + 6.

2014

Infinite families of linear binary nested completely regular codes with covering radius ρ equal to 3 and 4 are constructed. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter D = 3 or 4 are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive.

2011

Let (P, L, I) be a partial linear space and X ⊆ P ∪ L. Let us denote by (X)I = x∈X {y : yIx} and by [X] = (X)I ∪ X. With this terminology a partial linear space (P, L, I) is said to admit a (1, ≤ k)-identifying code if and only if the sets [X] are mutually different for all X ⊆ P ∪ L with |X| ≤ k. In this paper we give a characterization of k-regular partial linear spaces admitting a (1, ≤ k)-identifying code. Equivalently, we give a characterization of k-regular bipartite graphs of girth at least six admitting a (1, ≤ k)-identifying code. Moreover, we present a family of k-regular partial linear spaces on 2(k − 1) 2 + k points and 2(k − 1) 2 + k lines whose incidence graphs do not admit a (1, ≤ k)-identifying code. Finally, we show that the smallest (k; 6)-graphs known up to now for k − 1 not a prime power admit a (1

Electronic Journal of Graph Theory and Applications, 2021

Let G be a graph with |V (G)| vertices and ψ : V (G) −→ {1, 2, 3, • • • , |V (G)|} be a bijective function. The weight of a vertex v ∈ V (G) under ψ is w ψ (v) = u∈N (v) ψ(u). The function ψ is called a distance magic labeling of G, if w ψ (v) is a constant for every v ∈ V (G). The function ψ is called an (a, d)-distance antimagic labeling of G, if the set of vertex weights is a, a + d, a + 2d,. .. , a + (|V (G)| − 1)d. A graph that admits a distance magic (resp. an (a, d)distance antimagic) labeling is called distance magic (resp. (a, d)-distance antimagic). In this paper, we characterize distance magic 2-regular graphs and (a, d)-distance antimagic some classes of 2-regular graphs.

Designs, Codes and …, 1999

For strongly regular graphs with adjacency matrix A, we look at the binary codes generated by A and A + I. We determine these codes for some families of graphs, we pay attention to the relation between the codes of switching equivalent graphs and, with the exception of two parameter sets, we generate by computer the codes of all known strongly regular graphs on fewer than 45 vertices.

2020

Let G = (V, E) be a graph and f : V → Z2 be a vertex labeling. Define f+ : E → Z2 by f+(uv) = f(u) + f(v). For each i ∈ Z2, define vf (i) = |{v ∈ V : f(v) = i}| and ef (i) = ∣∣{e ∈ E : f+(e) = i}∣∣. If k ∈ N, then the vertex labeling f is a k-friendly labeling if |vf (0)− vf (1)| ≤ k. The k-friendly index of a graph G, denoted by ωk(G), is given by ωk(G) = max{|ef (0)− ef (1)| : f is a k-friendly labeling}. In this paper, we present the concepts k-friendly labeling and kfriendly index, and gave elementary results on the k-friendly index of paths, cycles, complete graphs, stars, lexicographic product of empty graphs with paths and cycles, planar grids, uniform n-star split graph, the graph SS(n, r), the graph CS(n, r), gear graphs, and sunlet graphs. Mathematics Subject Classification: 05C15 Keyword: Friendly Index, k-friendly labeling, friendly labeling, k-friendly index, paths, cycles 1This research is supported in part by the Rural Engineering and Technology Center of Negros Orien...

Discrete Mathematics, 2007

We show that every nonstable graph (possibly infinite) with bounded degree admits a set V \{x} as an identifying code. We know some infinite graphs for which only its vertex set itself is an identifying code, but these graphs have only vertices of infinite degree.

Annals of Combinatorics, 2016

A resolving set for a graph Γ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of Γ is the smallest size of a resolving set for Γ. Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distanceregular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs, but also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.

IEEE Transactions on Information Theory, 1998

We investigate a new class of codes for the optimal covering of vertices in an undirected graph G such that any vertex in G can be uniquely identified by examining the vertices that cover it. We define a ball of radius t centered on a vertex v to be the set of vertices in G that are at distance at most t from v: The vertex v is then said to cover itself and every other vertex in the ball with center v: Our formal problem statement is as follows: Given an undirected graph G and an integer t 1, find a (minimal) set C of vertices such that every vertex in G belongs to a unique set of balls of radius t centered at the vertices in C: The set of vertices thus obtained constitutes a code for vertex identification. We first develop topology-independent bounds on the size of C: We then develop methods for constructing C for several specific topologies such as binary cubes, nonbinary cubes, and trees. We also describe the identification of sets of vertices using covering codes that uniquely identify single vertices. We develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords. Finally, we describe an application of the theory developed in this paper to fault diagnosis of multiprocessor systems.

The Electronic Journal of Combinatorics, 2005

In this paper the problem of constructing graphs having a (1; ')-identifying code of small cardinality is addressed. It is known that the cardinality of such a code is bounded by ' 2 log ' logn . Here we construct graphs on n vertices having a( 1; ')-identifying code of cardinality O '4 logn for all ' 2. We derive

Journal of Combinatorial Theory, Series A, 2011

Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and with the same intersection numbers for any edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and derive some characterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem for distance-regular graphs is proved, so giving a quasi-spectral characterization of edgedistance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.

Designs, Codes and Cryptography, 2014

The hulls of codes from the row span over F p , for any prime p, of incidence matrices of connected k-regular graphs are examined, and the dimension of the hull is given in terms of the dimension of the row span of A + k I over F p , where A is an adjacency matrix for the graph. If p = 2, for most classes of connected regular graphs with some further form of symmetry, it was shown by Dankelmann et al. (Des. Codes Cryptogr. 2012) that the hull is either {0} or has minimum weight at least 2k − 2. Here we show that if the graph is strongly regular with parameter set (n, k, λ, μ), then, unless k is even and μ is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k − 2, and we construct words of low weight in the hull; if k is even and μ is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a k-regular graph, k ≥ 3, that has an -cycle for some ≥ 3, the binary hull is shown to be non-trivial with minimum weight at most 2 (k − 2). Properties of the p-ary hulls are also established.

Doklady Mathematics, 2010

Journal of Combinatorial Theory, Series A, 2011

Distance-regular graphs have been a key concept in Algebraic Combinatorics and have given place to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study 'almost distance-regular graphs'. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called m-walk-regularity. Another studied concept is that of m-partial distance-regularity, or informally, distance-regularity up to distance m. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of ( , m)-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem.

Discussiones Mathematicae Graph Theory, 2019

A (1, ≤ )-identifying code in a digraph D is a subset C of vertices of D such that all distinct subsets of vertices of cardinality at most have distinct closed in-neighbourhoods within C. In this paper, we give some sufficient conditions for a digraph of minimum in-degree δ -≥ 1 to admit a (1, ≤ )identifying code for ∈ {δ -, δ -+ 1}. As a corollary, we obtain the result by Laihonen that states that a graph of minimum degree δ ≥ 2 and girth at least 7 admits a (1, ≤ δ)-identifying code. Moreover, we prove that every 1-in-regular digraph has a (1, ≤ 2)-identifying code if and only if the girth of the digraph is at least 5. We also characterize all the 2-in-regular digraphs admitting a (1, ≤ )-identifying code for ∈ {2, 3}.