The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let $H_n$ be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of $H_n$ due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of $H_n$ by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to $H_n$ .

For a graph G, the resistance distance $r_G(x,y)$ is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index $R^{\ast }(G)={\sum }_{\{x,y\}\subset V(G)}{d}_G(x)d_G(y)r_G(x,y)$ , where $d_G(x)$ is the degree of vertex x, and $V(G)$ denotes the vertex set of G. L. Feng et al. obtained the element in $Cact(n;t)$ with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on $R^{\ast }(G)$ , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively.

For a t-uniform hypergraph H = (V(H), E(H)), the Estrada index EE(H) of H is defined as Ei is an element of[n] e theta i, where theta 1, ... , theta n are the eigevalues of the adjacency matrix of H. In this paper, the extremal t-uniform unicylic hypergraph which has maximum Estrada index are characterized.

$SLEE$ has various applications in a large variety of problems. The signless Laplacian Estrada index of a hypergraph H is defined as $SLEE\left(H\right)={\sum }_{i=1}^n{e}^{{\lambda }_i\left(Q\right)}$, where ${\lambda }_1\left(Q\right),{\lambda }_2\left(Q\right),\dots ,{\lambda }_n\left(Q\right)$ are the eigenvalues of the signless Laplacian matrix of H. In this paper, we characterize the unique r-uniform unicyclic hypergraphs with maximum and minimum $SLEE$.