Given a group \(G\), we define the power graph \(P(G)\) as follows: the vertices are the elements of \(G\) and two vertices \(x\) and \(y\) are joined by an edge if \(\langle x\rangle \subseteq \langle y \rangle\) or \(\langle y\rangle \subseteq \langle x \rangle\). Obviously, the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. In the present paper, among other results, we will find the number of spanning trees of the power graph associated with specific finite groups. We also determine, up to isomorphism, the structure of a finite group \(G\) whose power graph has exactly \(n\) spanning trees, for \(n < 5^{3}\). Finally, we show that the alternating group \(A_5\) is uniquely determined by the tree-number of its power graph among all finite simple groups.

Let \(G\) be a graph of order \(n\). The number of positive eigenvalues of \(G\) is called the positive inertia index of \(G\) and denoted by \(p(G)\). The minimum number of complete multipartite subgraphs in any complete multipartite graph edge decomposition of graph \(G\), in which the edge-induced subgraph of each edge subset of the decomposition is a complete multipartite graph, is denoted by \(\epsilon(G)\). In this paper, we prove \(\epsilon(G) \geq p(G)\) for any graph \(G\). Especially, if \(\epsilon(G) = 2\), then \(p(G) = 1\). We also characterize the graph \(G\) with \(p(G) = n – 2\).

The distance spectral gap of a connected graph is defined as the difference between its first and second distance eigenvalues. In this note, the unique \(n\)-vertex trees with minimal and maximal distance spectral gaps, and the unique \(n\)-vertex unicyclic graph with minimal distance spectral gap are determined.

A simple graph \(G = (V, E)\) admits an \(H\)-covering if every edge in \(E\) belongs to at least one subgraph of \(G\) isomorphic to a given graph \(H\). An \((a, d)\)-\(H\)-antimagic labeling of \(G\) admitting an \(H\)-covering is a bijective function \(f : V \cup E \rightarrow \{1, 2, \ldots, |V| + |E|\}\) such that, for all subgraphs \(H’\) of \(G\) isomorphic to \(H\), the \(H’\)-weights, \(wt(H’) = \sum_{v \in V(H’)} f(v) + \sum_{e \in E(H’)} f(e)\), constitute an arithmetic progression with the initial term \(a\) and the common difference \(d\). Such a labeling is called super if \(f(V) = \{1, 2, \ldots, |V|\}\). In this paper, we study the existence of super \((a, d)\)-\(H\)-antimagic labelings for graph operation \(G ^ H\), where \(G\) is a (super) \((b, d^*)\)-edge-antimagic total graph and \(H\) is a connected graph of order at least \(3\).

This article proves that the square of a Halin graph \(G\) with \(\Delta(G) = 5\) has the chromatic number \(6\). This gives a positive answer to an open problem in [Y. Wang, Distance two labelling of Halin graphs, Ars Combin. 114 (2014), 331–343].

There are many rectangular arrays whose \(n^{th}\) column is the \(n\)-fold convolution of the \(0^{th}\) column in combinatorics. For this type of rectangular arrays, we prove a formula for evaluating the determinant of certain submatrices, which was conjectured by Hoggatt and Bicknell. Our result unifies the determinant evaluation of submatrices of the rectangular arrays consisting of binomial coefficients, multinomial coefficients, Fibonacci numbers, Catalan numbers, generalized Catalan and Motzkin numbers.

In this paper, we obtain the following upper bounds for the largest Laplacian graph eigenvalue: \[\mu \leq \max\limits_{i} \left\{\sqrt{ 2d_i (m_i + d_i) + n – 2d_i – 2 \sum\limits_{j:j\sim i}{ |N_i \cap N_j|}} \right\}\] where \(d_i\) and \(m_i\) are the degree of vertex \(i\) and the average degree of vertex \(i\), respectively; \(|N_i \cap N_j|\) is the number of common neighbors of vertices \(i\) and \(j\). We also compare this bound with some known upper bounds.

A three-colored digraph \(D\) is primitive if and only if there exist nonnegative integers \(h\), \(k\), and \(v\) with \(h+k+v > 0\) such that for each pair \((i, j)\) of vertices there is an \((h, k, v)\)-walk in \(D\) from \(i\) to \(j\). The exponent of the primitive three-colored digraph \(D\) is defined to be the smallest value of \(h + k + v\) over all such \(h\), \(k\), and \(v\). In this paper, a class of special primitive three-colored digraphs with \(n\) vertices, consisting of one \(n\)-cycle and two \((n-1)\)-cycles, are considered. For the case \(a = c – 1\), some primitive conditions, the tight upper bound on the exponents, and the characterization of extremal three-colored digraphs are given.

Skew-quasi-cyclic codes over a finite field are viewed as skew-cyclic codes on a noncommutative ring of matrices over a finite field. This point of view gives a new construction of skew-quasi-cyclic codes. Let \(\mathbb{F}_q\) be the Galois field with \(q\) elements and \(\theta\) be an automorphism of \(\mathbb{F}_q\). We propose an approach to consider the relationship between left ideals in \(M_l(\mathbb{F}_q)[X, \theta]/(X^s – 1)\) and skew-quasi-cyclic codes of length \(ls\) and index \(l\) over \(\mathbb{F}_q\), under \(\theta\), which we denote by \(\theta\)-SQC codes (or SQC codes for short when there is no ambiguity). We introduce the construction of SQC codes from the reversible divisors of \(X^s – 1\) in \(M_l(\mathbb{F}_q)[X, \theta]\). In addition, we give an algorithm to search for the generator polynomials of general SQC codes.

In this paper, we investigate the concepts of \(k\)-limited packing and \(k\)-tuple domination in graphs and give several bounds on the size of them. These bounds involve many well-known parameters of graphs. Also, we establish a connection between these concepts that implies some new results in this area. Finally, we improve many bounds in the literature.