Euclidean graph distance matrices of generalizations of the star graph

Jaklič, Gašper; Modic, Jolanda

doi:10.1016/j.amc.2013.12.158

Cited by 20 publications

(35 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, a graph G admits a quadratic embedding (in this case we say that G is of QE class) if and only if QEC(G) ≤ 0. Thus, our study is closely related to the so-called Euclidean distance geometry [6,10,11,13]. Moreover, it is noteworthy that QEC(G) ≤ 0 is equivalent to the positive definiteness of the Q-matrix Q = [q d (i,j) ] for all 0 ≤ q ≤ 1.…”

Section: Introductionmentioning

confidence: 90%

Determining finite connected graphs along the quadratic embedding constants of paths

Baskoro

Obata

2021

EJGTA

View full text Add to dashboard Cite

The QE constant of a finite connected graph G, denoted by QEC(G), is by definition the maximum of the quadratic function associated to the distance matrix on a certain sphere of codimension two. We prove that the QE constants of paths P n form a strictly increasing sequence converging to −1/2. Then we formulate the problem of determining all the graphs G satisfying QEC(P n ) ≤ QEC(G) < QEC(P n+1 ). The answer is given for n = 2 and n = 3 by exploiting forbidden subgraphs for QEC(G) < −1/2 and the explicit QE constants of star products of the complete graphs.

show abstract

Section: Introductionmentioning

confidence: 90%

Determining finite connected graphs along the quadratic embedding constants of paths

Baskoro

Obata

2021

EJGTA

View full text Add to dashboard Cite

show abstract

“…Let D be the distance matrix of a graph G on n vertices with n ≥ 3. Let S(D) be the set of (f = [x i ], λ, µ) satisfying (10), (11) and (13). Then QEC(G) coincides with the maximum of λ appearing in S(D).…”

Section: Methods Of Lagrange Multipliersmentioning

confidence: 99%

“…Noteworthy characteristic properties of the distance matrices of trees are derived by Balaji-Bapat [2], see also Bapat [3,Chapter www.ejgta.org . More recently, Jaklič-Modic [11,12,13] determine some classes of graphs of QE class (in our terminology) and their explicit embeddings in Euclidean spaces. Second, the Q-matrix of a graph, the entrywise exponential of the distance matrix, has become important in relation to the q-deformation of spectral distributions of graphs, see e.g., Hora-Obata [9].…”

Section: Introductionmentioning

confidence: 99%

Distance matrices and quadratic embedding of graphs

Obata

Zakiyyah

2018

EJGTA

View full text Add to dashboard Cite

A connected graph is said to be of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently, if the distance matrix is conditionally negative definite. Several criteria for a graph to be of QE class are derived from the point of view of graph operations. For a quantitative criterion the QE constant is introduced and concrete examples are shown with explicit calculation. If the distance matrix admits a constant row sum, the QE constant coincides with the second largest eigenvalue of the distance matrix. The QE constants are determined for all graphs on n vertices with n ≤ 5, among which two are not of QE class.

show abstract

“…1). First four of them are the star graph S 4 , the path graph P 4 , the cycle graph C 4 and the complete graph K 4 , respectively, which are EDM-graphs (see [9,10]). Therefore we only need to consider the last two graphs, G …”

Section: The Smallest Nedm-graphsmentioning

confidence: 99%

“…Path and cycles were analysed in [9]. Star graphs and their generalizations were considered in [6,10]. Some results on Cartesian products of EDM-graphs are also known (see [11]).…”

Section: Introductionmentioning

confidence: 99%

On minimal forbidden subgraphs for the class of EDM-graphs

Jaklič

Modic

2014

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

In this paper, a relation between graph distance matrices and Euclidean distance matrices (EDM) is considered. Graphs, for which the distance matrix is not an EDM (NEDMgraphs), are studied. All simple connected non-isomorphic graphs on n ≤ 8 nodes are analysed and a characterization of the smallest NEDM-graphs, i.e., the minimal forbidden subgraphs, is given. It is proven that bipartite graphs and some subdivisions of the smallest NEDM-graphs are NEDM-graphs, too.

show abstract

Euclidean graph distance matrices of generalizations of the star graph

Cited by 20 publications

References 12 publications

Determining finite connected graphs along the quadratic embedding constants of paths

Determining finite connected graphs along the quadratic embedding constants of paths

Distance matrices and quadratic embedding of graphs

On minimal forbidden subgraphs for the class of EDM-graphs

Contact Info

Product

Resources

About