Some unicyclic graphs determined by the signless Laplacian permanental polynomial

• Published in: Journal of applied mathematics & computing Vol. 70; no. 4; pp. 2857 - 2878
• Main Authors: Khan, Aqib, Panigrahi, Pratima, Panda, Swarup Kumar
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2024; Springer Nature B.V
• Subjects: Apexes; Computational Mathematics and Numerical Analysis; Graph theory; Graphical representations; Graphs; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Matrix; Original Research; Permutations; Polynomials; Theory of Computation; Signless Laplacian permanental polynomial; 05C31; Unicyclic graph; Permanent; Pendant star; 05C50; Star degree; 15A15
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-024-02082-8

The permanent of a square matrix M = ( m ij ) k × k is p e r ( M ) = ∑ σ ∏ i = 1 k m i σ ( i ) , where the sum is taken over all permutations σ of the set { 1 , 2 , … , k } . For a simple connected graph G , its signless Laplacian matrix Q ( G ) is D ( G ) + A ( G ) , where D ( G ) is the degree diagonal matrix and A ( G ) is the adjacency matrix of G . The signless Laplacian permanental polynomial of G is ψ ( Q ( G ) ; x ) = p e r ( x I - Q ( G ) ) . In this paper, we give the representation of a graph with two types of degrees, d ( > 1 ) and 1, in terms of its signless Laplacian permanental polynomial. The unicyclic graphs discussed here are UC ( r ,  d ), where the unique cycle is C r with all whose vertices have degree d ( > 2 ) , and the remaining are degree 1 vertices. We show that UC ( r ,  d ) is determined uniquely by its signless Laplacian permanental polynomial for r = 3 , 4 , 5 , 6 and for any d . We also prove that the unicyclic graph obtained from UC ( r ,  d ) by making d - 1 new vertices adjacent to a pendant vertex, is also determined uniquely by its signless Laplacian permanental polynomial, for r = 4 , 5 . Finally, we show that among all connected graphs, UC ( r , 3), for every r ≥ 3 , is determined uniquely by its signless Laplacian permanental polynomial.