arXiv subjects

Raju Nandi

Publications and source records attributed to Raju Nandi.

4 recordsLinked to original sources

arXiv2025

Quadratic Embedding Constants of Cartesian Products and Joins of Graphs

The quadratic embedding constant (QEC) of a finite, simple, connected graph originated from the classical work of Schoenberg [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938] on Euclidean distance geometry. In this article, we study the QEC of graphs in terms of two graph operations: the Cartesian product and the join of graphs. We derive a general formula for the QEC of the join of an arbitrary graph with a regular graph and with a complete multipartite graph. As an application of these results, we explicitly compute the QEC for several classes of graphs and provide new examples of graphs of QE class. We also establish a lower bound for the quadratic embedding constant of the Cartesian product of two arbitrary connected graphs. Furthermore, as an extremal case, we derive concise formulas for the quadratic embedding constants of the Cartesian product of an arbitrary graph G with a complete graph and with a complete bipartite graph, expressed in terms of $\qec(G)$.

math.CO↗

arXiv2023

Matrices with simple symmetric digraphs and their group inverses

A new class of simple symmetric digraphs called $\mathcal{D}$ is defined and studied here. Any digraph in $\mathcal{D}$ has the property that each non-pendant vertex is adjacent to at least one pendant vertex. A graph theoretical description for the entries of the group inverse of a real square matrix with any digraph belonging to this class is given. We classify all the real square matrices $A$ such that the digraphs associated with $A$ and $A^{\#}$ both are in $\mathcal{D}$, that is, the digraph related to $A$ is either a corona or a star digraph.

math.CO↗

arXiv2023

Quadratic embedding constants of graphs: Bounds and distance spectra

The quadratic embedding constant (QEC) of a finite, simple, connected graph $G$ is the maximum of the quadratic form of the distance matrix of $G$ on the subset of the unit sphere orthogonal to the all-ones vector. The study of these QECs was motivated by the classical work of Schoenberg on quadratic embedding of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In this article, we provide sharp upper and lower bounds for the QEC of trees. We next explore the relation between distance spectra and quadratic embedding constants of graphs - and show two further results: $(i)$ We show that the quadratic embedding constant of a graph is zero if and only if its second largest distance eigenvalue is zero. $(ii)$ We identify a new subclass of nonsingular graphs whose QEC is the second largest distance eigenvalue. Finally, we show that the QEC of the cluster of an arbitrary graph $G$ with either a complete or star graph can be computed in terms of the QEC of $G$. As an application of this result, we provide new families of examples of graphs of QE class.

math.CO↗

arXiv2023

Group Inverses of Weighted Trees

Let $(G,w)$ be an undirected weighted graph. The group inverse of $(G,w)$ is the weighted graph with the adjacency matrix $A^{\#}$, where $A$ is the adjacency matrix of $(G,w)$. We study the group inverse of singular weighted trees. It is shown that if $(T,w)$ is a singular weighted tree, then $T^{\#}$ is again a tree, if and only if $T$ is a star tree, which in turn, holds if and only if $T^{\#}$ is graph isomorphic to $T$. A new class $\mathbb{T}$ of weighted trees, is introduced and studied here. It is shown that the group inverse of the adjacency matrix of a positively weighted tree in $\mathbb{T}$, is signature similar to a non-negative matrix.

math.CO↗