Degree-Based Weighted Adjacency Matrices: Spectra, Integrality, and Edge Deletion Effects

Abstract: The article presents weighted adjacency spectrum of complete multipartite graphs, characterize its families with three distinct eigenvalues and identifies integral matrices. Also, we observe that for almost all weighted matrices, the energy and the spectral radius of a complete graph decreases upon edge deletion, thereby correcting and refining earlier published results in [Bilal and Munir, Int. J. Quantum Chem. (2024)]. Furthermore, we give counter examples related to $ISI$ energy decrease of regular tripartite graph by edge deletion and give its correct $ISI$ spectrum and $ISI$ energy and settle an open problem related to $ISI$ energy change of the multipartite graph. Also, we calculate the weighted adjacency spectrum of crown multipartite graph and discuss its integral spectral weighted spectrum.

• The paper derives exact spectra for degree-weighted adjacency matrices of complete multipartite and crown graphs using equitable partitions, reducing integrality to whether the relevant weight value φ(d,d) is an integer.
• The paper establishes a ratio-based condition under which deleting an edge lowers spectral radius and energy in complete graphs, while computational evidence supports broader monotonicity conjectures.
• The paper corrects prior ISI-energy claims, showing that edge deletion can increase energy in regular multipartite graphs and providing counterexamples for complete graphs and tripartite graphs.

Overview

This paper by Rather and Ganie (2603.08895) studies the general extended (weighted) adjacency matrix $\dot{A}(G) = A_\phi(G)$, whose off-diagonal entries are given by a symmetric degree-based weight function $\phi(d_u, d_v)$ for adjacent vertices $u \sim v$. This framework unifies a broad family of degree-based matrices: the adjacency matrix $A$ ($\phi = 1$), the inverse sum indeg matrix $ISI$ ($\phi(x,y) = xy/(x+y)$), the arithmetic-geometric matrix $AG$, the geometric-arithmetic matrix $GA$, the first and second Zagreb matrices $M_1$ and $\phi(d_u, d_v)$0, the atom-bond connectivity matrix $\phi(d_u, d_v)$1, the Randić matrix $\phi(d_u, d_v)$2, the Sombor matrix $\phi(d_u, d_v)$3, and its reciprocal variant $\phi(d_u, d_v)$4. The paper makes three principal contributions: an exact spectral decomposition of $\phi(d_u, d_v)$5 for complete multipartite graphs, together with characterizations of three-distinct-eigenvalue families and integrality; a correction of erroneous results in Bilal and Munir's study of ISI energy change under edge deletion (2603.08895); and new spectra for crown multipartite graphs.

Weighted spectra of complete multipartite graphs

The central structural result is an exact description of the spectrum of $\phi(d_u, d_v)$6 for a complete $\phi(d_u, d_v)$7-partite graph $\phi(d_u, d_v)$8. Writing $\phi(d_u, d_v)$9 for the common degree within part $u \sim v$0, the authors prove that

$u \sim v$1

where $u \sim v$2 is the $u \sim v$3 quotient matrix with zero diagonal and entries $u \sim v$4 for $u \sim v$5. The proof proceeds by exhibiting $u \sim v$6 linearly independent null vectors supported on sums-zero subspaces within each part, then showing that the partition into parts is equitable, so that the remaining $u \sim v$7 eigenvalues coincide with those of the quotient matrix. The argument relies only on symmetry of $u \sim v$8 and constancy of degrees within parts, so it applies to every weight function in the family.

Two corollaries follow directly. First, for $u \sim v$9 and $A$0, the matrix $A$1 has exactly three distinct eigenvalues if and only if the multipartite graph is regular ($A$2). The converse direction is proved via a spectral decomposition argument: since $A$3 is symmetric with zero trace and only two distinct eigenvalues, its Perron eigenvector must be flat, forcing all off-diagonal entries of $A$4 to be equal and hence all part sizes equal. The authors explicitly note the necessity of the hypotheses: for $A$5 the bipartite spectrum $A$6 already yields three distinct eigenvalues without regularity, and for $A$7 (i.e., $A$8) the eigenvalue $A$9 disappears entirely. Second, for the regular case $\phi = 1$0 with degree $\phi = 1$1,

$\phi = 1$2

so the spectrum is integral exactly when $\phi = 1$3. Evaluating this criterion for standard weights shows that the adjacency, $\phi = 1$4, $\phi = 1$5, first Zagreb, and second Zagreb spectra are always integral, whereas the ISI spectrum is integral if and only if $\phi = 1$6 is even. These criteria reduce integrality questions to elementary arithmetic on the degree, avoiding any eigenvalue computation.

Energy change under edge deletion in complete graphs

The energy $\phi = 1$7 of $\phi = 1$8 equals $\phi = 1$9, and deleting an edge $ISI$0 yields a graph whose weighted energy is

$ISI$1

where $ISI$2 and $ISI$3. Comparing these expressions gives a clean threshold: both the spectral radius and the energy of $ISI$4 decrease upon edge deletion whenever

$ISI$5

The practical value of this criterion is that it depends only on the ratio of two evaluations of $ISI$6, not on any spectral computation.

Applying it to the ISI weight gives $ISI$7, so the ISI energy of $ISI$8 does not increase upon edge deletion — contradicting Theorem 2 of Bilal and Munir. The authors give an explicit numerical witness at $ISI$9: the ISI spectrum of $\phi(x,y) = xy/(x+y)$0 is $\phi(x,y) = xy/(x+y)$1 with energy $\phi(x,y) = xy/(x+y)$2, while that of $\phi(x,y) = xy/(x+y)$3 is $\phi(x,y) = xy/(x+y)$4 with energy $\phi(x,y) = xy/(x+y)$5. A computational survey across all listed weight functions at comparable orders confirms that the complete-graph energy decreases upon edge deletion in every case examined, leading the authors to state as a conjecture that $\phi(x,y) = xy/(x+y)$6 holds for almost all functions $\phi(x,y) = xy/(x+y)$7. The conjecture is supported by computation rather than proved, and no class of exceptional weight functions is identified.

Correction of the tripartite ISI results

The paper then corrects two further claims from Bilal and Munir concerning the ISI energy of the regular tripartite graph $\phi(x,y) = xy/(x+y)$8. For $\phi(x,y) = xy/(x+y)$9 with $AG$0 joining vertices in the first two parts, the authors derive the full ISI spectrum using an equitable five-cell partition (the two endpoints of the deleted edge, the remainders of parts one and two, and the third part). The eigenvalue $AG$1 occurs with multiplicity exactly $AG$2; two simple eigenvalues are given in closed form,

$AG$3

and the remaining three eigenvalues are the roots of an explicit cubic polynomial $AG$4. Because the cubic cannot be solved in closed form, no closed formula for the resulting energy is available; the energy is expressed as $AG$5, where $AG$6 denotes the energy of the explicit $AG$7 restriction matrix.

Numerical evaluation shows that the formula in Theorem 6 of Bilal and Munir substantially understates the true energy — for example, at order $AG$8 the corrected value is $AG$9 against the previously published $GA$0. More significantly, comparing $GA$1 with $GA$2 yields counterexamples to Theorem 7 of that paper for every $GA$3: the energy increases upon edge deletion (e.g., $GA$4 at $GA$5), while the claim happens to hold only for $GA$6. Since $GA$7 for an $GA$8-regular graph, the authors transfer Akbari–Ghorbani–Oboudi's edge-addition result for adjacency energy to obtain a theorem covering all regular $GA$9-partite graphs with $M_1$0: deleting an edge strictly increases the ISI energy. This resolves the open problem posed in the earlier paper concerning energy change of multipartite graphs under edge deletion, and it demonstrates that the monotonicity behavior of ISI energy differs qualitatively between complete graphs (decrease) and regular multipartite graphs (increase).

Crown multipartite graphs and the star-plus-edge case

For the crown graph $M_1$1 (obtained from $M_1$2 by deleting a perfect matching), the weighted spectrum is $M_1$3 with $M_1$4, giving energy $M_1$5. The analysis extends to the $M_1$6-partite crown $M_1$7 of order $M_1$8: the spectrum consists of $M_1$9 with multiplicity $\phi(d_u, d_v)$00, $\phi(d_u, d_v)$01 with multiplicity $\phi(d_u, d_v)$02, $\phi(d_u, d_v)$03 simple, and $\phi(d_u, d_v)$04 with multiplicity $\phi(d_u, d_v)$05 (the last recovered from the trace condition), yielding energy $\phi(d_u, d_v)$06. As in the complete multipartite case, the spectrum is integral if and only if $\phi(d_u, d_v)$07 is an integer, so the ISI spectrum of the crown is integral exactly when $\phi(d_u, d_v)$08 is even, while the adjacency, $\phi(d_u, d_v)$09, $\phi(d_u, d_v)$10, $\phi(d_u, d_v)$11, and $\phi(d_u, d_v)$12 spectra are always integral.

Finally, the paper notes that Theorem 10 of Bilal and Munir concerning ISI energy of the star under edge deletion is ill-posed: deleting an edge from $\phi(d_u, d_v)$13 disconnects the graph, and omitting the isolated vertex compares energies of graphs of different orders. The authors instead compare the star $\phi(d_u, d_v)$14 with the unicyclic graph $\phi(d_u, d_v)$15 obtained by adding one edge. The three nonzero nontrivial eigenvalues of $\phi(d_u, d_v)$16 are roots of a cubic with no closed-form solution, so the comparison rests on computation: across tested orders from $\phi(d_u, d_v)$17 to $\phi(d_u, d_v)$18, $\phi(d_u, d_v)$19 throughout (e.g., $\phi(d_u, d_v)$20 versus $\phi(d_u, d_v)$21 at $\phi(d_u, d_v)$22), indicating that the ISI energy of the star increases upon edge addition. No analytic proof of this monotonicity is provided.

Limitations and open questions

Several limitations are stated plainly in the paper. The three-distinct-eigenvalue characterization excludes the bipartite case $\phi(d_u, d_v)$23 and the case $\phi(d_u, d_v)$24, where the equivalence fails. The conjecture that complete-graph energy decreases under edge deletion for almost all weight functions is empirical, with no characterization of potential exceptions. The tripartite ISI energy involves a cubic whose roots lack closed form, precluding an exact energy formula and leaving the analytic confirmation of the computed inequalities open. Similarly, the star-versus-$\phi(d_u, d_v)$25 comparison is computational only. The concluding section acknowledges that determining whether energy increases or decreases upon edge deletion for a general weighted matrix remains difficult and unresolved.

Conclusion

The paper delivers an exact equitable-partition-based spectral theory for degree-weighted adjacency matrices of complete multipartite and crown multipartite graphs, with clean integrality criteria reducing to whether $\phi(d_u, d_v)$26 is an integer. Its corrective contributions are substantive: a ratio-based threshold governing energy and spectral radius change for complete graphs, explicit counterexamples showing that prior published ISI energy claims fail for $\phi(d_u, d_v)$27 ($\phi(d_u, d_v)$28 sufficing) and for regular tripartite graphs with $\phi(d_u, d_v)$29, and a resolution of the open multipartite energy-change problem via the regularity reduction to known adjacency-energy results. The work also highlights how much of the remaining territory — closed-form energies involving irreducible cubics, and the "almost all" weight-function conjecture — still depends on computation rather than proof.