For a finite group G, the power graph P(G) is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x and y are adjacent if and only if xⁱ = y or y^j = x, for 2 ≤ i, j ≤ n. In this paper, we obtain the distance Laplacian spectrum of power graphs of finite groups such as cyclic groups, dihedral groups, dicyclic groups, abelian groups and elementary abelian p groups. Moreover, we find the largest and second smallest distance Laplacian eigenvalue of power graphs of such groups.

The Merrifield-Simmons index σ is the total number of independent vertex sets (including the empty set) of the graph G. The Wiener index W is the sum of distances in all unordered pairs of vertices of G. We construct some new graphs satisfying σ > W and W > σ, respectively. In particular, infinite graphs satisfying W > σ are invented with graphs with diameter 2 and infinite ones satisfying σ > W are discovered with so-called universally diametrical graphs.

Given a graph $G$ and a positive integer $k$, define the Gallai-Ramsey number to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain exact values of the Gallai-Ramsey numbers for the union of two stars in many cases and bounds in other cases. This work represents the first class of disconnected graphs to be considered as the desired monochromatic subgraph.

For $0 \leq \alpha < 1$, the $\alpha$-spectral radius of an $r$-uniform hypergraph $G$ is the spectral radius of $\mathcal{A}_{\alpha}(G)=\alpha \mathcal{D}(G)+(1-\alpha)\mathcal{A}(G)$, where $\mathcal{D}(G)$ and $\mathcal{A}(G)$ are the diagonal tensor of degrees and adjacency tensor of $G$, respectively. In this paper, we show the perturbation of $\alpha$-spectral radius by contracting an edge. Then we determine the unique unicyclic hypergraph with the maximum $\alpha$-spectral radius among all $r$-uniform unicyclic hypergraphs with fixed diameter. We also determine the unique unicyclic hypergraph with the maximum $\alpha$-spectral radius among all $r$-uniform unicyclic hypergraphs with given number of pendant edges.

The skewness of a graph G, denoted by sk(G), is the minimum number of edges in G whose removal results in a planar graph. It is an important parameter that measures how close a graph is to planarity, and it is complementary, and computationally equivalent, to the Maximum Planar Subgraph Problem. For any connected graph G on p vertices and q edges with girth g, one can easily verify that sk(G) ≥ π(G), where π(G) = ?q - g/(g-2) (p - 2)?, and the graph G is said to be π-skew if equality holds. The concept of π-skew was first proposed by G. L. Chia and C. L. Lee. The π-skew graphs with girth 3 are precisely the graphs that contain a triangulation as a spanning subgraph. The purpose of this paper is to explore the properties of π-skew graphs. Some families of π-skew graphs are obtained by applying these properties, including join of two graphs, complete multipartite graphs and Cartesian product of two graphs. We also discuss the threshold for the existence of a spanning triangulation. Among other results some sufficient conditions regarding the regularity and size of a graph, which ensure a spanning triangulation, are given.