Matrix analysis
Topological graph theory
Characteristic polynomials of some graph coverings
Discrete Mathematics
On graphs with the maximum number of spanning trees
Proceedings of the seventh international conference on Random structures and algorithms
Transformations of a graph increasing its Laplacian polynomial and number of spanning trees
European Journal of Combinatorics
The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs
Discrete Applied Mathematics
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Given a graph G with vertex set V(G)=V and edge set E(G)=E, let G^l be the line graph and G^c the complement of G. Let G^0 be the graph with V(G^0)=V and with no edges, G^1 the complete graph with the vertex set V, G^+=G and G^-=G^c. Let B(G) (B^c(G)) be the graph with the vertex set V@?E such that (v,e) is an edge in B(G) (resp., in B^c(G)) if and only if v@?V, e@?E and vertex v is incident (resp., not incident) to edge e in G. Given x,y,z@?{0,1,+,-}, the xyz-transformationG^x^y^zofG is the graph with the vertex set V(G^x^y^z)=V@?E and the edge set E(G^x^y^z)=E(G^x)@?E((G^l)^y)@?E(W), where W=B(G) if z=+, W=B^c(G) if z=-, W is the graph with V(W)=V@?E and with no edges if z=0, and W is the complete bipartite graph with parts V and E if z=1. In this paper we obtain the Laplacian characteristic polynomials and some other Laplacian parameters of every xyz-transformation of an r-regular graph G in terms of |V|, r, and the Laplacian spectrum of G.