Let $ G $ be a graph of order $ n $ with $ m $ edges. Let $ \tilde{L}(G) $ be the Laplacian atom-bond connectivity (ABC)-matrix of $ G $. The eigenvalues of $ \tilde{L}(G) $ are the Laplacian ABC-eigenvalues of $ G $, and its largest eigenvalue is called the Laplacian ABC-spectral radius of $ G $. In this paper, we present some new upper and lower bounds for the Laplacian ABC-spectral radius, and we characterize the extremal graphs attaining these bounds. We also present an upper bound for the second smallest Laplacian ABC-eigenvalue and characterize the extremal graphs. We determine the Laplacian ABC-eigenvalues of the join of two regular graphs and the join of a regular graph with the union of two regular graphs, in terms of the Laplacian eigenvalues of the component graphs.
Citation: Hilal A. Ganie, Amal Alsaluli. New results on the Laplacian ABC-eigenvalues of a graph[J]. AIMS Mathematics, 2026, 11(8): 25768-25794. doi: 10.3934/math.20261032
Let $ G $ be a graph of order $ n $ with $ m $ edges. Let $ \tilde{L}(G) $ be the Laplacian atom-bond connectivity (ABC)-matrix of $ G $. The eigenvalues of $ \tilde{L}(G) $ are the Laplacian ABC-eigenvalues of $ G $, and its largest eigenvalue is called the Laplacian ABC-spectral radius of $ G $. In this paper, we present some new upper and lower bounds for the Laplacian ABC-spectral radius, and we characterize the extremal graphs attaining these bounds. We also present an upper bound for the second smallest Laplacian ABC-eigenvalue and characterize the extremal graphs. We determine the Laplacian ABC-eigenvalues of the join of two regular graphs and the join of a regular graph with the union of two regular graphs, in terms of the Laplacian eigenvalues of the component graphs.
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Hilal A. Ganie, Amal Alsaluli. New results on the Laplacian ABC-eigenvalues of a graph[J]. AIMS Mathematics, 2026, 11(8): 25768-25794. doi: 10.3934/math.20261032
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