Permanental Polynomials Research Articles

On the Roots of (Signless) Laplacian Permanental Polynomials of Graphs

On the Roots of (Signless) Laplacian Permanental Polynomials of Graphs

A study on determination of some graphs by Laplacian and signless Laplacian permanental polynomials

The permanent of an n × n matrix is defined as where the sum is taken over all permutations σ of The permanental polynomial of M, denoted by is where In is the identity matrix of order n. Let G be a simple undirected graph on n vertices and its Laplacian and signless Laplacian matrices be L(G) and Q(G) respectively. The permanental polynomials and are called the Laplacian permanental polynomial and signless Laplacian permanental polynomial of G respectively. A graph G is said to be determined by its (signless) Laplacian permanental polynomial if all the graphs having the same (signless) Laplacian permanental polynomial with G are isomorphic to G. A graph G is said to be combinedly determined by its Laplacian and signless Laplacian permanental polynomials if all the graphs having (i) the same Laplacian permanental polynomial as and (ii) the same signless Laplacian permanental polynomial as are isomorphic to G. In this article we investigate the determination of some graphs, namely, star, wheel, friendship graphs and a particular kind of caterpillar graph (whose all r non-pendant vertices have the same degree n) by their Laplacian and signless Laplacian permanental polynomials. We show that a kind of caterpillar graphs (for ), wheel graph (up to 7 vertices) and friendship graph (up to 7 vertices) are determined by their (signless) Laplacian permanental polynomials. Further we prove that all friendship graphs and wheel graphs are combinedly determined by their Laplacian and signless Laplacian permanental polynomials.

Permanents of almost regular complete bipartite graphs

Let G be a graph, and let A ( G ) be the adjacency matrix of G. The computation of permanent of A ( G ) is #p-complete. Computing permanent of A ( G ) is of great interest in quantum chemistry, statistical physics, among other disciplines. In this paper, we characterize the ordering of permanents of adjacency matrices of all graphs obtained from regular complete bipartite graph K p , p by deleting six edges. As an application, we show that all graphs with a perfect matching obtained from K p , p with six edges deleted are determined by their permanental polynomials.

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.

On the matching and permanental polynomials of graphs

The matching polynomials and permanental polynomials have been investigated extensively, but no investigations focus on the relations between them. In this paper, we establish first two equalities on the signless matching polynomial and signless permanental polynomials. Then we show that the number of perfect matchings of a graph is the mean value of the permanents of skew-adjacency matrices. Moreover, we find that the matching polynomial of a graph is the expectation of skew-permanental polynomials of orientation graphs. Finally, we derive that the variance of the permanents of skew-adjacency matrices of orientation graphs can be expressed in terms of the number of matching alternating cycles in a graph.

Graphs determined by the (signless) Laplacian permanental polynomials

ABSTRACT Characterizing which graphs are determined by their (signless) Laplacian permanental polynomials is an interesting problem. In this paper, we prove that paths, cycles and lollipop graphs are determined by their Laplacian permanental polynomials as well as their signless Laplacian permanental polynomials, respectively.

On the adjacency matrix of a complex unit gain graph

ABSTRACT A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same.

The cycle (circuit) polynomial of a graph with double and triple weights of edges and cycles

Farrell introduced the general class of graph polynomials which he called the family polynomials, or F -polynomials, of graphs. One of these is the cycle , or circuit , polynomial. This polynomial is in turn a common generalization of the characteristic, permanental, and matching polynomials of a graph, as well as a wide variety of statistical-mechanical partition functions, such as were earlier known. Herein, we specially derive weighted generalizations of the characteristic and permanental polynomials requiring for calculation thereof to assign double (res. triple) weights to all Sachs subgraphs of a graph. To elaborate an analytical method of calculation, we extend our earlier differential-operator approach which is now employing operator matrices derived from the adjacency matrix. Some theorematic results are obtained.

On the (Signless) Laplacian Permanental Polynomials of Graphs

Let G be a graph, and let L(G) and Q(G) denote respectively the Laplacian matrix and the signless Laplacian matrix of G. 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 give combinatorial expressions for the first five coefficients of the (signless) Laplacian permanental polynomial. The characterizing properties of the (signless) Laplacian permanental polynomial are investigated and some graphs determined by the (signless) Laplacian permanental polynomial are presented. Furthermore, we compute the (signless) Laplacian permanental polynomials for all graphs on at most 10 vertices, and count the number of such graphs for which there is another graph with the same (signless) Laplacian permanental polynomial.

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 ) .

The graphs whose permanental polynomials are symmetric

The graphs whose permanental polynomials are symmetric

Highly unique network descriptors based on the roots of the permanental polynomial

In this paper, we examine the zeros of permanental polynomials as highly unique network descriptors. We employ exhaustively generated networks and demonstrate that our defined graph measures based on the moduli of the zeros of permanental polynomials are quite efficient when distinguishing graphs structurally. In this work, we continue with a line of research that relates to the search of almost complete graph invariants. These highly unique network measures may serve as a powerful tool for tackling graph isomorphism.

Computing the permanental polynomials of graphs

Let M be an n × n matrix with entries mij (i,j=1,2,…,n). The permanent of M is defined to beper(M)=∑σ∏i=1nmiσ(i),where the sum is taken over all permutations σ of {1,2,…,n}. The permanental polynomial of M is defined by per(xIn−M), where In is the identity matrix of size n. In this paper, we give recursive formulas for computing permanental polynomials of the Laplacian matrix and the signless Laplacian matrix of a graph, respectively.

Graphs whose characteristic and permanental polynomials have coefficients of the same magnitude

Let ϕ(G,x)=∑i=0naixn−i and π(G,x)=∑i=0nbixn−i denote the characteristic polynomial and permanental polynomial of a graph G. In this paper, we consider the family G of graphs that have corresponding coefficients of the same magnitude, i.e., |ai|=|bi| for i=0,1,…,n. We prove that the graphs in this family are planar graphs. To characterize the structures of the considered graphs, we introduce the plane ear decomposition. With the help of the plane ear decomposition, we show the characterizations of the bipartite and non-bipartite graphs in this family.

On the Permanental Polynomials of Matrices

An \(m\times n\,\{0,1\}\)-matrix \(A\) is said to be totally convertible if there exists a matrix \(B\) obtained from \(A\) by changing some 1’s in \(A\) to \(-1\)’s such that for any submatrix \(A^{\prime }\) of \(A\) of order \(m\), the corresponding submatrix \(B^{\prime }\) of \(B\) satisfies \(\mathrm{per}(xI-A^{\prime })=\det (xI-B^{\prime })\). In this paper, motivated by the well-known Polya’s problem, our object is to characterize those totally convertible matrices. Associate a matrix \(A\) with a bipartite graph \(G^{*}_A\). We first prove that a square matrix \(A\) is totally convertible if and only if \(G^{*}_A\) is Pfaffian, and then we generalize this result to an \(m\times n\)\(\{0,1\}\)-matrix. Moreover, the characterization of a totally convertible matrix provides an equivalent condition to compute the permanental polynomial of a bipartite graph by the characteristic polynomial of the skew adjacency matrix of its orientation graph. As applications, we give some explicit expressions of the permanental polynomials of two totally convertible matrices by the technique of Pfaffian orientation.

The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.

The Permanental Polynomials of Subdivision Graphs

The Permanental Polynomials of Subdivision Graphs

Characterizing properties of permanental polynomials of lollipop graphs

A lollipop graph, denoted by, is a graph obtained by appending a cycle to a pendant vertex of a path. It is known that the lollipop graphs are determined by their spectra [Haemers WH, Liu X, Zhang Y. Spectral characterizations of lollipop graphs. Linear Algebra Appl. 2008;428:2415–2423; Boulet R, Jouve B. The lollipop graph is determined by its spectrum. Electron. J. Combin. 2008;15:#R74]. In this paper, we consider whether the lollipop graphs can be determined by the permanental polynomial. We show that () and are determined by their permanental polynomials when restricted to connected graphs, and in general () cannot be determined by their permanental polynomials even when restricted to connected graphs. In particular, we find disconnected copermanental mates of for and connected copermanental mates of,,, and for, respectively.

On the characterizing properties of the permanental polynomials of graphs

Let G be a graph and A(G) the adjacency matrix of G. The permanental polynomial of G is defined as π(G,x)≜per(xI-A(G)). If two graphs G and H have the same permanental polynomial, then G is called a per-cospectral mate of H. A graph G is said to be characterized by its permanental polynomial if all the per-cospectral mates of G are isomorphic to G. It is shown that complete graphs, stars, regular complete bipartite graphs, and odd cycles are characterized by their permanental polynomials. We prove that in general the permanental polynomial cannot characterize the paths and even cycles. In particular, for each l⩾1 and m⩾2, we can find non-isomorphic per-cospectral mates of P4l+3 and C4m, respectively. When we restrict our consideration to connected graphs, both the paths and even cycles C4l+2 are characterized by their permanental polynomials.

Computing the permanental polynomials of bipartite graphs by Pfaffian orientation

The permanental polynomial of a graph G is π(G,x)≜per(xI−A(G)). From the result that a bipartite graph G admits an orientation Ge such that every cycle is oddly oriented if and only if it contains no even subdivision of K2,3, Yan and Zhang showed that the permanental polynomial of such a bipartite graph G can be expressed as the characteristic polynomial of the skew adjacency matrix A(Ge). In this note we first prove that this equality holds only if the bipartite graph G contains no even subdivision of K2,3. Then we prove that such bipartite graphs are planar. Unexpectedly, we mainly show that a 2-connected bipartite graph contains no even subdivision of K2,3 if and only if it is planar 1-cycle resonant. This implies that each cycle is oddly oriented in any Pfaffian orientation of a 2-connected bipartite graph containing no even subdivision of K2,3. Accordingly, we give a way to compute the permanental polynomials of such graphs by Pfaffian orientation.