Spectral Analysis of Eccentricity Matrix for One-Point-Corona of Graphs

Selvaganesh Lavanya (Indian Institute of Technology (BHU))

The corona product is a prominent graph operation known for its distinct structural features and has been the focus of extensive research. Over the time, several variations of the corona product have been introduced and analyzed. In this talk, we explore the spectral characteristics of the eccentricity matrix for one such variation of corona product.

Let \(G\) and \(H\) be connected graphs with \(m\) and \(n\) vertices, respectively, where \(H\) is rooted at a designated vertex \(z\). The one-point-corona, denoted as \(G \circ_z H\), is constructed by taking one copy of \(H\) for each vertex of \(G\) and joining the root vertex \(z\) from each copy of \(H\) to the corresponding vertex in \(G\) with an edge.

We study the properties of eccentricity matrix, \(\varepsilon\), for one-point-corona \(G \circ_z H\). %First, we determine the conditions under which \(\varepsilon(G \circ_z H)\) is irreducible by proving the connectedness of the direct product of two graphs, where one graph includes a self-loop.

Assuming \(G\) is self-centered, we examine the \(\varepsilon\)-spectrum of \(G \circ_z H\) , identify classes of \(\varepsilon\)-cospectral graphs, and analyze the extremal graphs with respect to their spectral radius.

In our exploration of extremal graphs based on their \(\varepsilon\)-spectral radius, we achieve a more generalized result by arranging the graphs \(G \circ_z H\) in a linear order of their spectral radius, assuming \(G\) is self-centered and \(H\) is any rooted graph with a fixed eccentricity for its root vertex. This is a joint work with Ms. Smrati Pandey.

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