In this paper, we determine the third largest and the fourth largest numbers of independent sets among all trees of order \(n\). Moreover, we determine the \(k\)-th largest numbers of independent sets among all forests of order \(n\), where \(k \geq 2\). Besides, we characterize those extremal graphs achieving these values.

For a set \(\mathcal{P}\) of permutations, the sign-imbalance of \(\mathcal{P}\) is the difference between the numbers of even and odd permutations in \(\mathcal{P}\).In this paper, we determine the sign-imbalances of two classes of alternating permutations ,one is the Alternating permutations avoiding a pattern of length three and the other is the Alternating permutations of genus \(0\)
The sign-imbalance of the former involves Catalan and Fine numbers, and that of the latter is always \(\pm 1\).Meanwhile, we give a simpler proof of Dulucq and Simion’s result on the number of alternating permutations of genus \(0\).

A survivable path \((W, P)\) between a pair of vertices \(x_i, x_j\) in an undirected simple graph \(G\) is an ordered pair of edge-disjoint simple paths consisting of a working path \(W = x_i, \ldots, x_j\) a protection path \(P = x_i, \ldots, x_j\).An optimal set of survivable paths in graph \(G\) corresponds to a set of mesh-restored lightpaths defined on an optical network that minimizes the number of used optical channels.In this paper, we present new properties of the working paths, which are contained in an optimal set of survivable paths in \(G\).

We describe the global behavior of the nonnegative equilibrium points of the difference equation

\[x_{n+1} = \frac{ax_{n -p}}{b+c \prod\limits_{i=0}^{k} x_{n-(2i+1)}},n=0,1,\ldots,\]

where \(k,p \in \mathbb{N}\), parameters \(a,b,c\) and initial conditions are nonnegative real numbers.

Let \(\mathcal{T}_{n,n-4}\) be the set of trees on \(n\) vertices with diameter \(n-4\). In this paper, we determine the unique tree which has the minimal Laplacian spectral radius among all trees in \(\mathcal{T}_{n,n-4}\).
This work is related to that of Yuan [The minimal spectral radius of graphs of order n with diameter \(n – 4\), Linear Algebra Appl. \(428(2008)2840-2851]\), which determined the graph with minimal spectral radius among all the graphs of order \(n\) with diameter \(n-4\). We can observe that the extremal tree on the Laplacian spectral radius is different from that on the spectral radius.

We introduce the notion of vague Lie sub-superalgebras (resp. vague ideals) and present some of their properties. We investigate the properties of vague Lie sub-superalgebras and vague ideals under homomorphisms of Lie superalgebras.We introduce the concept of vague bracket product and establish its characterizations. We also introduce the notions of solvable vague ideals and nilpotent vague ideals of Lie superalgebras and present the corresponding theorems parallel to Lie superalgebras.

The atom-bond connectivity (ABC) index of a graph \(G\) is defined in mathematical chemistry as\(\mathrm{ABC}(G) = \sum_{uv \in E(G)} \sqrt{\frac{d_u +d_v-2}{ d_u d_v}},\) where \(E(G)\) is the edge set of \(G\) and \(d_u\) is the degree of vertex \(u\) in \(G\).In this paper, we determine the unique graphs with the largest and the second largest ABC indices, respectively, in the class of unicyclic graphs on \(2m\) vertices with perfect matchings.

Let \(\Delta\) be one of the dual polar spaces \(\mathrm{DQ}(8, q)\), \(\mathrm{DQ}^-(7,q)\), and let \(e: \Delta \to \Sigma\) denote the spin-embedding of \(\Delta\). We show that \(e(\Delta)\) is a two-intersection set of the projective space \(\Sigma\). Moreover, if \(\Delta \cong \mathrm{DQ}^-(7,q)\), then \(e(\Delta)\) is a \((q^3 + 1)\)-tight set of a nonsingular hyperbolic quadric \(\mathrm{Q}^+(7,q^2)\) of \(\Sigma \cong PG(7,q^2)\). This \((q^2 + 1)\)-tight set gives rise to more examples of \((q^3 + 1)\)-tight sets of hyperbolic quadrics by a procedure called field-reduction.All the above examples of two-intersection sets and \((q^3 + 1)\)-tight sets give rise to two-weight codes and strongly regular graphs.

Let \(G = (V, E)\) be a simple undirected graph. An independent set is a subset \(S \subseteq V\) such that no two vertices in \(S\) are adjacent. A maximal independent set is an independent set that is not a proper subset of any other independent set.
In this paper, we study the problem of determining the fourth largest number of maximal independent sets among all trees and forests. Extremal graphs achieving these values are also given.

From differential operators and the generating functions of Bernoulli and Euler polynomials, we derive some new theorems on Bernoulli and Euler numbers. By using integral formulae and arithmetical properties relating to the Bernoulli and Euler polynomials, we obtain new identities on Bernoulli and Euler numbers. Finally, we give some new properties on Bernoulli and Euler numbers arising from the \(p\)-adic integrals on \(\mathbb{Z}_p\).