Abstract:
In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on $n$ vertices with girth $g$ ($n,g$ being fixed), which graph minimizes the Laplacian spectral radius? We prove that the graph $U_{n,g}$ (defined in Section 1) uniquely minimizes the Laplacian spectral radius for $n\geq 2g-1$ when $g$ is even and for $n\geq 3g-1$ when $g$ is odd.

Abstract:
Let $G$ be an unicyclic graph of order $n$ and let $Q_G(x)= det(xI-Q(G))={matrix} \sum_{i=1}^n (-1)^i \varphi_i x^{n-i}{matrix}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\mathcal{G}(n,m)$. $\mathcal{G}(n,m)$ denotes all n-vertex unicyclic graphs with matching number $m$. We characterize the graphs which minimize all the signless Laplacian coefficients in the set $\mathcal{G}(n,m)$ with odd (resp. even) girth. Moreover, we find the extremal graphs which have minimal signless Laplacian coefficients in the set $\mathcal{G}(n)$ of all $n$-vertex unicyclic graphs with odd (resp. even) girth.

Abstract:
Let be the characteristic polynomial of the Laplacian matrix of a graph of order . In this paper, we give four transforms on graphs that decrease all Laplacian coefficients and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs with vertices and pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with vertices and pendent vertices.

Abstract:
We introduce a new operation on a class of graphs with the property that the Laplacian eigenvalues of the input and output graphs are related. Based on this operation, we obtain a family of order (square root of n) noncospectral unicyclic graphs on n vertices with the same Laplacian energy.

Abstract:
In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order $n$ with girth and the matching number $\beta\ge \frac{3g}{2}$. Moreover, we characterize all extremal graphs which attain the lower bound.

Abstract:
For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$,where $q_1, q_2, \dots, q_n$are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$ and then determine the unique unicyclic graph with maximum $SLEE$ among the unicyclic graphs on $n$ vertices with given diameter.

Abstract:
Let be the set of all unicyclic graphs with vertices and cycle length . For any , consists of the (unique) cycle (say ) of length and a certain number of trees attached to the vertices of having (in total) edges. If there are at most two trees attached to the vertices of , where is even, we identify in the class of unicyclic graphs those graphs whose Laplacian spectral radii are minimal. 1. Introduction Following [1], let be a simple connected graph on vertices and edges (so is its order and is its size). For , the degree of is denoted by . Let be the maximum degree of . For two vertices and ？？( ) in , the distance between and , denoted by , is the number of edges in a shortest path joining and . Let be the set of all unicyclic graphs on vertices and the cycle length . So, if , then consists of the (unique) cycle (say ) of length and a certain number of trees attached to vertices of having (in total) edges. If the cycle length of a unicyclic graph is even (odd), we call it an even (odd) unicyclic graph. We may assume that vertices of are or for short only (ordered in a natural way around , say, in the clockwise direction). For each , let be a rooted tree (with as its root) attached to . Then, for each , we can write . If , for each , is a path , whose root is a vertex of minimum degree, then we write . Let be the adjacency matrix of and the diagonal matrix. Then the Laplacian matrix of is . Since is real symmetric and positive semidefinite, its eigenvalues are nonnegative real numbers. For a graph , we denote by the largest eigenvalue of and call it the Laplacian spectral radius. The investigation on the Laplacian spectral radius of graphs is an important topic in the theory of graph spectra. Since 1980s, there are several monographs and a lot of research papers published continually (see [2–7]). Recently, the problem concerning graphs with maximal or minimal Laplacian spectral radius of a given class of graphs has been studied by many authors. Guo [8] determined the first four graphs with the largest Laplacian spectral radius among all unicyclic graphs on vertices. In this paper, for any , if there are at most two trees attached to vertices of , where is even, we characterize in the class of even unicyclic graphs those graphs whose Laplacian spectral radii are minimal. 2. Main Results and Proofs Let be the principal submatrix obtained from by deleting the corresponding row and column of . Generally, let be the principal submatrix obtained from by deleting the corresponding rows and columns of all vertices of . For any square matrix , denote by the

The nullity of a
graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper
we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs
with extremal nullity.

Abstract:
We identify graphs with the maximal Laplacian spectral radius among all unicyclic graphs with vertices and diameter . 1. Introduction Following [1], let be a simple undirected graph on vertices and edges (so is its order and is its size). For , or denotes the degree of and denotes the set of all neighbors of vertex . A pendant vertex is a vertex of degree 1 and a pendant edge is an edge incident with a pendant vertex. Let . For two vertices and ( ), the distance between and is the number of edges in the shortest path joining and . The diameter of a graph is the maximum distance between any two vertices of . Let ( ) be a path of with (unless ). If , , then we call an internal path of ; if and , then we call a pendant path of ; if the subgraph induced by in is itself, that is, , then we call an induced path. Obviously, the shortest path between any two distinct vertices of is an induced path. We will use , to denote the graph obtained from by deleting a vertex , or an edge , respectively (this notation is naturally extended if more than one vertex, or edge, is deleted). Denote by and the cycle and the path with vertices, respectively. We call a unicyclic graph if , where is the number of vertices and is the number of edges. We will use to denote the sets of all unicyclic graphs with vertices and diameter . Let be a graph of order obtained from the cycle by attaching pendant edges and a path of length at one vertex of the cycle, and a path of length to another nonadjacent vertex of the cycle respectively, where . Let be the Laplacian matrix, where is the diagonal matrix and is the adjacency matrix. The matrix is real symmetric and positive semidefinite; the eigenvalues of can be arranged as , where the largest eigenvalue is called the Laplacian spectral radius of . The investigation on the Laplacian spectral radius of graphs is an important topic in the theory of graph spectra. Recently, the problem concerning graphs with maximal Laplacian spectral radius of a given class of graphs has been studied extensively. Li et al. [2] determined those graphs which maximized Laplacian spectral radius among all bipartite graphs with (edge-) connectivity at most and characterized graphs of order with cut-edges, having Laplacian spectral radius equal to . X. L. Zhang and H. P. Zhang [3] studied the largest Laplacian spectral radius of the bipartite graphs with vertices and cut edges and the bicyclic bipartite graphs, respectively. The Laplacian spectral radius of unicyclic graphs has been studied by many authors (see [4–6]). Liu et al. [7] determined the graphs with the

Abstract:
The trees, respectively unicyclic graphs, on $n$ vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the $n$-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the $n$-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian permanent are characterized. Consequently, the $n$-vertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent are determined.