Characterizing properties of permanental polynomials of lollipop graphs

Liu, Shunyi; Zhang, Heping

doi:10.1080/03081087.2013.779271

Cited by 15 publications

(6 citation statements)

References 14 publications

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“…So they stated that the per-spectrum seems a little better than the spectrum when it comes to distinguishing graphs which are not trees. Liu and Zhang [23,24] investigated paths, stars, cycles and lollipop graphs which are DPS. They found that graphs determined by the characteristic polynomial are not necessarily determined by the permanental polynomial.…”

Section: Introductionmentioning

confidence: 99%

Per-spectral characterizations of graphs with extremal per-nullity

Zhang

2015

Linear Algebra and its Applications

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Section: Introductionmentioning

confidence: 99%

Per-spectral characterizations of graphs with extremal per-nullity

Zhang

2015

Linear Algebra and its Applications

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“…It was found that the coefficients and roots of π(A(G), x) encode the structural information of a (chemical) graph G (see, e.g., [13,14]). Characterization of graphs by the permanental polynomial has been investigated, see [15][16][17][18][19]. The Laplacian permanental polynomial of a graph was first considered by Merris et al [11], and the signless Laplacian permanental polynomial was first studied by Faria [20].…”

Section: Introductionmentioning

confidence: 99%

Generalized Permanental Polynomials of Graphs

Liu

2019

Symmetry

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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 ) .

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“…In particular, they showed that a bipartite graph has no real permanental roots except possibly zero. Additionally, several papers have been published on graphs uniquely determined by their permanental spectra, see [16,17,23,30], among others.…”

Section: Introductionmentioning

confidence: 99%