The Characterizing Properties of (Signless) Laplacian Permanental Polynomials of Almost Complete Graphs

Let G be a graph with n vertices, and let L G and Q G denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L G (respectively, Q G ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.

The extremal pentagon-chain polymers with respect to permanental sum

The permanental sum of a graph G can be defined as the sum of absolute value of coefficients of permanental polynomial of G. It is closely related to stability of structure of a graph, and its computing complexity is #P-complete. Pentagon-chain polymers is an important type of organic polymers. In this paper, we determine the upper and lower bounds of permanental sum of pentagon-chain polymers, and the corresponding pentagon-chain polymers are also determined.

The Extremal Permanental Sum for a Quasi-Tree Graph

Let G be a graph and A(G) the adjacency matrix of G. The permanent of matrix (xI-A(G)) is called the permanental polynomial of G. The permanental sum of G is the sum of the absolute values of the coefficients of permanental polynomial of G. Computing the permanental sum is #p-complete. In this note, we prove the maximum value and the minimum value of permanental sum of quasi-tree graphs. And the corresponding extremal graphs are also determined. Furthermore,we also determine the graphs with the minimum permanental sum among quasi-tree graphs of order n and size m, where n-1≤m≤2n-3.

Generalized Permanental Polynomials of Graphs

The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to find a complete graph invariant, we introduce the generalized permanental polynomials of graphs. Let G be a graph with adjacency matrix A ( G ) and degree matrix D ( G ) . The generalized permanental polynomial of G is defined by P G ( x , μ ) = per ( x I − ( A ( G ) − μ D ( G ) ) ) . In this paper, we compute the generalized permanental polynomials for all graphs on at most 10 vertices, and we count the numbers of such graphs for which there is another graph with the same generalized permanental polynomial. The present data show that the generalized permanental polynomial is quite efficient for distinguishing graphs. Furthermore, we can write P G ( x , μ ) in the coefficient form ∑ i = 0 n c μ i ( G ) x n − i and obtain the combinatorial expressions for the first five coefficients c μ i ( G ) ( i = 0 , 1 , ⋯ , 4 ) of P G ( x , μ ) .