Per-spectral characterizations of graphs with extremal per-nullity

Constructing Matching Equivalent Graphs

“…Let G be a graph obtained by identifying a cycle 4 C to vertex u of ( ) 22 1, 1, 1 G n n n − − − , and let H be a graph obtained by attaching a path 2 P to a vertex of degree two in ( ) 22 , , G n n n , where 3 n ≥ (see Figure 2). Then G H and G H .…”

On the permanental nullity and matching number of graphs

“…By the definition of a Sachs graph, S(G) has three possible structures: a maximum matching, union of disjoint cycles, or union of some disjoint single edges and cycles. In [21], two elementary properties of per-nullity of graphs are introduced as follows.…”

“…Wu and H. Zhang [21]) Let G be a graph with n vertices and S(G) be a maximum Sachs subgraph of G. Then η per (G) = n − |V (S(G))|.…”

Generalized Permanental Polynomials of Graphs

“…It was found that the coefficients and roots of π(A(G), x) encode the structural information of a (chemical) graph G (see, e.g., [13,14]). Characterization of graphs by the permanental polynomial has been investigated, see [15][16][17][18][19]. The Laplacian permanental polynomial of a graph was first considered by Merris et al [11], and the signless Laplacian permanental polynomial was first studied by Faria [20].…”