Spectral properties of a new distance-based matrix

Articles in Press

Document Type : Original paper

Authors

Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, Iran

Abstract

In this paper, we introduce and analyze the spectral properties of the Graovac-Ghorbani matrix $\tilde{\mathcal{A}}$. We first calculate the Graovac-Ghorbani index for specific classes of structures, including nicely distance-balanced, vertex-transitive, and edge-transitive graphs. By defining the Laplacian matrix associated with $\tilde{\mathcal{A}}$, we prove that for any connected graph of order $n \ge 3$, the rank of this Laplacian matrix is exactly $n-1$. We establish sharp upper and lower bounds for the spectral radius of $\tilde{\mathcal{A}}$ in terms of graph parameters for various families, including trees, unicycle bipartite graphs, and general bipartite graphs, characterizing the extremal cases such as complete bipartite graphs. Furthermore, the spectral properties of strongly distance-balanced (SDB) graphs are investigated. A key finding of this study is the proof that the $\tilde{\mathcal{A}}$-eigenvalues of any bipartite graph are symmetric with respect to zero. However, we demonstrate that the converse does not hold, implying that a symmetric $\tilde{\mathcal{A}}$-spectrum does not necessarily guarantee the bipartiteness of the graph.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 04 May 2026

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