Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Farrugia, Alexander

doi:10.3390/sym13091663

Cited by 3 publications

(3 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , w n−1 may be actually deduced from PD(G) [2,8]. We now proceed to use Theorem 3.1 to interpret the trace of A k adj(xI − A) for k = 1 and k = 2:…”

Section: The Trace Of a K Adj(xi − A)mentioning

confidence: 99%

“…(If G 1 is the graph on two vertices joined by a single edge and G 2 is the graph on two vertices and no edges, then PD(G 1 ) = PD(G 2 ), so the case n = 2 is not included in the PRP.) While this problem has been solved in the affirmative for a variety of classes of graphs [3,5,8,18,[20][21][22][23], it is still an open problem in general. Schwenk [17] expresses his doubt that ϕ(G, x) can be recovered from PD(G) for all graphs, arguing that a large graph will eventually be uncovered which will turn out to be a counterexample to the PRP.…”

mentioning

confidence: 99%

“…We remark that graphs having less than n − 1 edges are necessarily disconnected. Results on the PRP for disconnected graphs were reported in [8,20]. A graph having n − 1 edges is either disconnected or a tree.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Recovering the characteristic polynomial of a graph from entries of the adjugate matrix

Farrugia

2022

ELA

View full text Add to dashboard Cite

The adjugate matrix of $G$, denoted by $\operatorname{adj}(G)$, is the adjugate of the matrix $x\mathbf{I}-\mathbf{A}$, where $\mathbf{A}$ is the adjacency matrix of $G$. The polynomial reconstruction problem (PRP) asks if the characteristic polynomial of a graph $G$ can always be recovered from the multiset $\operatorname{\mathcal{PD}}(G)$ containing the $n$ characteristic polynomials of the vertex-deleted subgraphs of $G$. Noting that the $n$ diagonal entries of $\operatorname{adj}(G)$ are precisely the elements of $\operatorname{\mathcal{PD}}(G)$, we investigate variants of the PRP in which multisets containing entries from $\operatorname{adj}(G)$ successfully reconstruct the characteristic polynomial of $G$. Furthermore, we interpret the entries off the diagonal of $\operatorname{adj}(G)$ in terms of characteristic polynomials of graphs, allowing us to solve versions of the PRP that utilize alternative multisets to $\operatorname{\mathcal{PD}}(G)$ containing polynomials related to characteristic polynomials of graphs, rather than entries from $\operatorname{adj}(G)$.

show abstract

“…. , w n−1 may be actually deduced from PD(G) [2,8]. We now proceed to use Theorem 3.1 to interpret the trace of A k adj(xI − A) for k = 1 and k = 2:…”

Section: The Trace Of a K Adj(xi − A)mentioning

confidence: 99%