Let $G$ be a connected graph with vertex set $V\left(G\right)$. The degree distance of $G$ is defined as $D^{\prime }\left(G\right)={\sum }_{\{u,v\}\subseteq V\left(G\right)}({\mathrm{deg}}_G\left(u\right)+{\mathrm{deg}}_G\left(v\right))d(u,v)$, where ${\mathrm{deg}}_G\left(u\right)$ is the degree of vertex $u$, and $d(u,v)$ denotes the distance between $u$ and $v$. Here we characterize $n$-vertex unicyclic graphs with girth $k$, having minimum and maximum degree distance, respectively. Furthermore, we prove that the graph $B_n$, obtained from two triangles linked by a path, is the unique graph having the maximum degree distance among bicyclic graphs, which resolves a recent conjecture of Tomescu.

For a bipartite graph G and a non-zero real α, we give bounds for the sum of the αth powers of the Laplacian eigenvalues of G using the sum of the squares of degrees, from which lower and upper bounds for the incidence energy, and lower bounds for the Kirchhoff index and the Laplacian Estrada index are deduced.

The integral circulant graph $X_n (D)$ has the vertex set $Z_n = \{0, 1,\ldots$, $n{-}1\}$ and vertices $a$ and $b$ are adjacent, if and only if $\gcd(a{-}b$, $n)\in D$, where $D = \{d_1,d_2, \ldots, d_k\}$ is a set of divisors of $n$. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper, we deal with the automorphism group of integral circulant graphs and investigate a problem proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. We determine the size and the structure of the automorphism group of the unitary Cayley graph $X_n (1)$ and the disconnected graph $X_n (d)$. In addition, based on the generalized formula for the number of common neighbors and the wreath product, we completely characterize the automorphism groups $Aut (X_n (1, p))$ for $n$ being a square-free number and $p$ a prime dividing $n$, and $Aut (X_n (1, p^k))$ for $n$ being a prime power.