On the rank of the distance matrix of graphs (original) (raw)

The inertia of distance matrices of some graphs

Discrete Mathematics, 2013

Let D denote the distance matrix of a connected graph G. The inertia of D is the triple of integers (n + (D), n 0 (D), n − (D)), where n + (D), n 0 (D), n − (D) denote the number of positive, 0, and negative eigenvalues of D, respectively. In this paper, we mainly give some graphs whose n + (D) is equal to 1 and get the inertia of their distance matrices.

Graphs with few distinct distance eigenvalues irrespective of the diameters

The Electronic Journal of Linear Algebra, 2015

The distance matrix of a simple connected graph GGG is D(G)=(dij)D(G)=(d_{ij})D(G)=(dij), where dijd_{ij}dij is the distance between iiith and jjjth vertices of GGG. The multiset of all eigenvalues of D(G)D(G)D(G) is known as the distance spectrum of GGG. Lin et al.(On the distance spectrum of graphs. \newblock {\em Linear Algebra Appl.}, 439:1662-1669, 2013) asked for existence of graphs other than strongly regular graphs and some complete kkk-partite graphs having exactly three distinct distance eigenvalues. In this paper some classes of graphs with arbitrary diameter and satisfying this property is constructed. For each kin4,5,ldots,11k\in \{4,5,\ldots,11\}kin4,5,ldots,11 families of graphs that contain graphs of each diameter grater than k−1k-1k−1 is constructed with the property that the distance matrix of each graph in the families has exactly kkk distinct eigenvalues. While making these constructions we have found the full distance spectrum of square of even cycles, square of hypercubes, corona of a transmission regular graph with $K_...

On The Graphs with Maximum Distance or k-Diameter

2000

The distance of a set of vertices is the sum of the distances between pairs of vertices in the set. We dene the k-diameter of a graph as the maximum distance of a set of k vertices; so the 2-diameter is the normal diameter and the n-diameter where n is the order is the distance of the graph. We complete

Realization of distance matrices by graphs of genus 1

arXiv: Combinatorics, 2020

Given a distance matrix DDD, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of DDD by a weighted graph. To this end, we first give a general result on realization by n−n-n−cycles and successively we mainly focus on graphs of genus 1, presenting an algorithm which determines when a distance matrix is realizable by such a kind of graph, and then, shows how to construct it.

Distance–regular graphs having the M -property

Linear and Multilinear Algebra, 2012

We analyze when the Moore-Penrose inverse of the combinatorial Laplacian of a distanceregular graph is an M -matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore-Penrose inverse of the combinatorial Laplacian of a distance-regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance-regular graph has the M -property. We prove that only distance-regular graphs with diameter up to three can have the M -property and we give a characterization of the graphs that satisfy the M -property in terms of their intersection array. Moreover, we exhaustively analyze strongly regular graphs having the M -property and we give some families of distance regular graphs with diameter three that satisfy the M -property. Roughly speaking, we prove that all distance-regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance-regular graphs with diameter three, and no distance-regular graphs with diameter greater than three, have the M -property. In addition, we conjecture that no primitive distance-regular graph with diameter three has the M -property.

On the distance spectrum of distance regular graphs

Linear Algebra and its Applications, 2015

The distance matrix of a simple graph G is D(G) = (d ij), where d ij is the distance between ith and jth vertices of G. The spectrum of the distance matrix is known as the distance spectrum or D-spectrum of G. A simple connected graph G is called distance regular if it is regular, and if for any two vertices x, y ∈ G at distance i, there are constant number of neighbors c i and b i of y at distance i − 1 and i + 1 from x respectively. In this paper we prove that distance regular graphs with diameter d have at most d + 1 distinct D-eigenvalues. We find an equitable partition and the corresponding quotient matrix of the distance regular graph which gives the distinct D-eigenvalues. We also prove that distance regular graphs satisfying b i = c d−1 have at most d 2 + 2 distinct D-eigenvalues. Applying these results we find the distance spectrum of some distance regular graphs including the well known Johnson graphs. Finally we also answer the questions asked by Lin et al. [16].

On some problems regarding distance-balanced graphs

2022

A graph Gamma\GammaGamma is said to be distance-balanced if for any edge uvuvuv of Gamma\GammaGamma, the number of vertices closer to uuu than to vvv is equal to the number of vertices closer to vvv than to uuu, and it is called nicely distance-balanced if in addition this number is independent of the chosen edge uvuvuv. A graph Gamma\GammaGamma is said to be strongly distance-balanced if for any edge uvuvuv of Gamma\GammaGamma and any integer kkk, the number of vertices at distance kkk from uuu and at distance k+1k+1k+1 from vvv is equal to the number of vertices at distance k+1k+1k+1 from uuu and at distance kkk from vvv. In this paper we answer an open problem posed by Kutnar and Miklavi\v{c} [European J. Combin. 39 (2014), 57-67] by constructing several infinite families of nonbipartite nicely distance-balanced graphs which are not strongly distance-balanced. We disprove a conjecture regarding characterization of strongly distance-balanced graphs posed by Balakrishnan et al. [European J. Combin. 30 (2009), 1048-1053] by pro...

On Certain Distance Graphs and related Applications

2019

Stimulated by the famous plane coloring problem Eggleton coined the term distance graph and studied widely the prime distance graphs. A prime distance graph (PDG) G(Z,D) is one whose vertex set V is the set of integers Z and the distance set D is a subset of the set of primes P . The edge set of G denoted E is the one whose elements (u, v) for any u, v ∈ V (G) are characterized by the property that d(u, v) ∈ D where d(u, v) = |u − v| . According to J.D.Laison, C. Starr and A. Walker a graph G is a PDG if there exists a 1-1 labeling f : V (G)→ Z such that for any two adjacent vertices u and v the integer |f(u)− f(v)| is a prime. Further they called such a labelling of V (G) a prime distance labelling (PDL) of G . In this paper we prove certain existence and non-existence results concerning PDG and PDL and study the relationship between them. We also discuss certain applications besides raising some open problems.

On distance-regular graphs with smallest eigenvalue at least -m

Journal of Combinatorial Theory, 2010

A non-complete geometric distance-regular graph is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for a fixed integer m⩾2m⩾2, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least −m, diameter at least three and intersection number c2⩾2c2⩾2.

Graph Families with Constant Distance Determinant

The Electronic Journal of Combinatorics, 2018

This paper introduces a new class of graphs, the clique paths (or the CP graphs), and shows that their distance determinant and distance inertia are independent of their structures. The CP graphs include the family of linear 222-trees. When a graph is attached to a CP graph, it is shown that the distance determinant and the distance inertia are also independent of the structure of the CP graph. Applications to the addressing problem proposed by Graham and Pollak in 1971 are given.