On Infinite Families of $k-$Almost Borderenergetic Line Graphs

The concept of graph energy, defined as the sum of the absolute values of the eigenvalues of a graph's adjacency matrix, plays a central role in spectral graph theory and mathematical chemistry. A graph \(G\) of order \(n\) is called \emph{borderenergetic} if its energy equals that of the complete graph \(K_n\), namely \(E(G)=2(n-1)\). A recent generalization introduced \emph{almost borderenergetic} graphs, whose energies deviate from \(2(n-1)\) by less than one. In this paper, we further extend this idea by defining a broader class of graphs, called \emph{\(k\)-almost borderenergetic graphs}, for which the deviation from \(2(n-1)\) satisfies \(k-1 < |E(G)-(2n-2)| \leq k\), where \(k\) is a positive integer.Within this framework, we construct and analyze three infinite families of line graphs derived from join operations:\[\Omega_2 = \{\mathcal{L}((rK_2)\nabla K_1\mid r \geq 3 )\}, \quad\Omega_3 = \{\mathcal{L}((rK_3)\nabla K_1)\mid r \geq 1 \}, \quad\text{and} \quad\Omega_4 = \{\mathcal{L}((rK_1)\nabla 2K_1\mid r \geq 2 )\}.\]We determine their adjacency spectra and show that the corresponding line graphs are \(k\)-almost borderenergetic for \(k=2\), \(k=3\) and \(k=3\), respectively. These results reveal that line graphs can exhibit controlled spectral deviations from borderenergetic graphs and that \(k\)-almost borderenergetic families form a rich and structured generalization of previously known energy classes.