Correction to: D-Integral, DQ-Integral and DL-Integral Generalized Double-Wheel Graphs (Bulletin of the Malaysian Mathematical Sciences Society, (2024), 47, 4, (118), 10.1007/s40840-024-01710-7)

The main results (Theorems 2.12 and 3.8) in the above-titled original article [1] were not sufficiently accurate, as the regularity of Km,m was mistakenly taken as m-1 instead of the correct value m. To ensure the accuracy and integrity of the paper, we would like to issue the following corrections to the original article. We emphasize that the proof method proposed in the original article is still valid. Two of the eigenvalues of the graph GDW(a,m,n)=aKm,m∇Cn in Theorem 2.4 of [1] were incorrectly stated, due to an error in taking the regularity of Km,m to be m-1 instead of m. Consequently, we restate Theorem 2.4 as follows. The distance spectrum of the generalized double-wheel graph GDW(a, m, n) consists of the eigenvalues -m-2, -2, m-2 and -2cos2πjn-2 with multiplicities a-1, a(2m-2), a and 1 respectively, where 1≤j≤n-1, the remaining two eigenvalues are (Formula presented.) Using the corrected Theorem 2.4 and the original proof method of [1], we can derive all D-integral generalized double-wheel graphs as follows. The generalized double-wheel graph GDW(a, m, n) is D-integral if and only if the ordered triple (a,m,n)∈S, where (Formula presented.) This same reason (assigning the regularity of Km,m as m-1 rather than m) led to incorrect eigenvalues in Theorem 3.2 of [1]. The correct version of Theorem 3.2 is stated as follows. The distance signless Laplacian spectrum of the generalized double-wheel graph GDW(a, m, n) consists of the eigenvalues 4(am-1)+n-2m, 4(am-1)+n-m, 4(am-1)+n and 2am+2n-6-2cos2πjn with multiplicities a-1, a(2m-2), a and 1 respectively, where 1≤j≤n-1, the remaining two eigenvalues are (Formula presented.) Using the corrected Theorem 3.2 and the original proof method of [1], we can derive all DQ-integral generalized double-wheel graphs as follows. The generalized double-wheel graph GDW(a, m, n) is DQ-integral if and only if the ordered triple (a,m,n)∈S, where (Formula presented.)