Distance matrices, dimension, and conference graphs

Type-II Matrices Attached to Conference Graphs

We determine the Nomura algebras of the type-II matrices belonging to the Bose-Mesner algebra of a conference graph. 1 Type-II Matrices and Nomura Algebras We say that an n × n matrix W with complex entries is type II if W (j, i)(W)(i, j) = 1 n for i, j = 1, . . . , n. So a type-II matrix is invertible and has no zero entry. We use I and J to denote the identity matrix and the matrix of all one...

The Steiner distance dimension of graphs

For a nonempty set S of vertices of a connected graph G, the Steiner distance d(S) of S is the minimum size among all connected subgraphs whose vertex set contains S. For an ordered set W = {Wl, W2,"', Wk} of vertices in a connected graph G and a vertex v of G, the Steiner representation s(vIW) of v with respect to W is the (2k I)-vector where d i1 ,i2, ... ,ij(V) is the Steiner distance d({V,W...

The Directed Distance Dimension of Oriented Graphs

For a vertex v of a connected oriented graph D and an ordered set W = {w1, w2, . . . , wk} of vertices of D, the (directed distance) representation of v with respect to W is the ordered k-tuple r(v ∣ ∣ W ) = (d(v,w1), d(v, w2), . . . , d(v, wk)), where d(v, wi) is the directed distance from v to wi. The set W is a resolving set for D if every two distinct vertices of D have distinct representat...

Ela on Euclidean Distance Matrices of Graphs

Received by the editors on November 13, 2011. Accepted for publication on July 8, 2013. Handling Editor: Bryan L. Shader. FMF and IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia, and IAM, University of Primorska, Slovenia ([email protected]). XLAB d.o.o., Pot za Brdom 100, 1125 Ljubljana, Slovenia, and FMF, University of Ljubljana, Slovenia ([email protected]...

On Euclidean distance matrices of graphs