Year: 2025
Type: article
Abstract: The generalized matrix of a graph G is defined as M(G)=A(G)−tD(G) (t∈R, and A(G) and D(G), respectively, denote the adjacency matrix and the degree matrix of G), and the generalized characteristic polynomial of G is merely the characteristic polynomial of M(G). Let Km,n be the complete bipartite graph. Then, the Km,n-complement of a subgraph G in Km,n is defined as the graph obtained by removing all edges of an isomorphic copy of G from Km,n. In this paper, by using a determinant expansion on the sum of two matrices (one of which is a diagonal matrix), a general method for computing the generalized characteristic polynomial of the Km,n-complement of a bipartite subgraph G is provided. Furthermore, when G is a graph with rank no more than 4, the explicit formula for the generalized characteristic polynomial of the Km,n-complements of G is given.
Source: Symmetry
DOI: 10.3390/sym17030328
Authors: Weiliang Zhao, Helin Gong
Institutions: Zhejiang Industry Polytechnic College, Shaoxing University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,896
Awards: G1806862494, G4134672041, G6478561256
OpenAlex ID: W4407837991