Some unicyclic graphs determined by the signless Laplacian permanental polynomial

The permanent of a square matrix \(M=(m_{ij})_{k\times k}\) $M={(m_{\mathit{ij}})}_{k\times k}$ is \( per(M)=\sum _\sigma \prod _{i=1}^{k} m_{i\sigma (i)}\) $per(M)={\sum }_{\sigma }{\prod }_{i=1}^k{m}_{i\sigma (i)}$ , where the sum is taken over all permutations \(\sigma \) $\sigma$ of the set \(\{1,2,\ldots ,k\}\) $\{1,2,\dots ,k\}$ . For a simple connected graph G, its signless Laplacian matrix Q(G) is \(D(G)+A(G)\) $D(G)+A(G)$ , 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 \(\psi (Q(G);x) = per(xI-Q(G))\) $\psi (Q(G);x)=per(xI-Q(G))$ . In this paper, we give the representation of a graph with two types of degrees, \(d(>1)\) $d(>1)$ and 1, in terms of its signless Laplacian permanental polynomial. The unicyclic graphs discussed here are UC(r, d), where the unique cycle is \(C_r\) $C_r$ with all whose vertices have degree \(d(>2)\) $d(>2)$ , and the remaining are degree 1 vertices. We show that UC(r, d) is determined uniquely by its signless Laplacian permanental polynomial for \(r=3,4,5,6\) $r=3,4,5,6$ and for any d. We also prove that the unicyclic graph obtained from UC(r, d) by making \(d-1\) $d-1$ new vertices adjacent to a pendant vertex, is also determined uniquely by its signless Laplacian permanental polynomial, for \(r=4,5\) $r=4,5$ . Finally, we show that among all connected graphs, UC(r, 3), for every \(r\ge 3\) $r\ge 3$ , is determined uniquely by its signless Laplacian permanental polynomial.