Abstract
On the Permanental Polynomial and Permanental Sum of Signed Graphs
Highlights
The permanent of an n × n real matrix A =, with i, j ∈ {1, 2, . . . , n}, is defined as per(A) = Πni=1aiτ(i)τ where the sum is taken over all permutations τ of {1, 2, . . . , n}
K=0 is called the characteristic polynomial of G, where I is the n × n identity matrix
The characteristic polynomial and the permanental polynomial are important among the well-studied graph polynomials
Summary
It was shown that the coefficients of the characteristic and permanental polynomials of graphs are related to graphs’ structures [3, 4, 7]. A linear graph (or a Sachs graph) is a graph in which each component is a single edge or a cycle. A linear subgraph of a graph G is termed as a subgraph whose components are cycles or single edges. Uk ⊆Gfor any k(1 ≤ k ≤ n), where s(Uk) is the number of negative edges in cycles of Uk, p(Uk) is the number of components of Uk, c(Uk) is the number of cycles of Uk and c−(Uk) is the number of negative cycles of Uk. we introduce the permanental polynomial of signed graph Gdefined as n π(G , x) = per(xI − A(G )) = skxn−k (7).
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