Using the companion matrices, we get more identities and Hessenberg matrices about Fibonacci and Tribonacci numbers.
By Fibonacci and Tribonacci numbers we can evaluate the determinants and permanents of some special Hessenberg matrices.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A function \(f: E(G) \rightarrow \{-1, 1\}\) is said to be a signed star dominating function of \(G\) if \(\sum_{e \in E_G(v)} f(e) \geq 1\) for every \(v \in V(G)\), where \(E_G(v) = \{uv \in E(G) | u \in V(G)\}\). The minimum of the values of \(\sum_{e \in E(G)} f(e)\), taken over all signed dominating functions \(f\) on \(G\), is called the signed star domination number of \(G\) and is denoted by \(\gamma_{SS}(G)\). In this paper, we prove that \(frac{n}{2}\leq \gamma_{SS}(T) \leq n-1\) for every tree \(T\) of order \(n\), and characterize all trees on \(n\) vertices with signed star domination number \(\frac{n}{2}\), \(\frac{n+1}{2}\), \(n-1\), or \(n-3\).

The concept of rainbow connection was introduced by Chartrand et al. in 2008. The rainbow connection number, \(rc(G)\), of a connected graph \(G = (V, E)\) is the minimum number of colors needed to color the edges of \(E\), so that each pair of vertices in \(V\) is connected by at least one path in which no two edges are assigned the same color. The rainbow vertex-connection number, \(rvc(G)\), is the vertex version of this problem. In this paper, we introduce mixed integer programming models for both versions of the problem. We show the validity of the proposed models and test their efficiency using a nonlinear programming solver.

A graph of order \(n\) is \(p\)-factor-critical, where \(p\) is an integer with the same parity as \(n\), if the removal of any set of \(p\) vertices results in a graph with a perfect matching. It is well known that a connected vertex-transitive graph is \(1\)-factor-critical if it has odd order and is \(2\)-factor-critical or elementary bipartite if it has even order. In this paper, we show that a connected non-bipartite vertex-transitive graph \(G\) with degree \(k \geq 6\) is \(p\)-factor-critical, where \(p\) is a positive integer less than \(k\) with the same parity as its order, if its girth is not less than the bigger one between \(6\) and \( \frac{k(p-1)+8}{2(k-2)}\).

In this paper, the completely regular endomorphisms of a split graph are investigated. We give necessary and sufficient conditions the completely regular endomorphisms of a split graph form a monoid.

In this paper, we interpret a generalized basic series as the generating function of two different combinatorial objects, viz., a restricted \(n\)-colour partition function, which we call a two-colour partition function, and a weighted lattice path function. This leads to infinitely many combinatorial identities. Our main result has the potential of yielding many Rogers-Ramanujan-MacMahon type combinatorial identities. This is illustrated by an example.

Let \(u\) and \(v\) be two vertices in a graph \(G\). We say vertex \(u\) dominates vertex \(v\) if \(N(v) \subseteq N(u) \cup \{u\}\). If \(u\) dominates \(v\) or \(v\) dominates \(u\), then \(u\) and \(v\) are comparable. The Dilworth number of a graph \(G\), denoted \(\text{Dil}(G)\), is the largest number of pairwise incomparable vertices in the graph \(G\). A graph \(G\) is called \(\{H_1, H_2, \ldots, H_k\}\)-free if \(G\) contains no induced subgraph isomorphic to any \(H_i\), \(1 \leq i \leq k\). A graph \(G\) is called an \(L_1\)-graph if, for each triple of vertices \(u\), \(v\), and \(w\) with \(d(u,v) = 2\) and \(w \in N(u) \cap N(v)\), \(d(u)+d(v) \geq |N(u) \cup N(v) \cup N(w)| – 1\). Let \(G\) be a \(k\) (\(k \geq 2\))-connected \(L_2\)-graph. If \(G\) is \(\{K_{1,5}, K_{1,5+e}\}\)-free and \(\text{Dil}(G) \leq 2k-1\), then \(G\) is Hamiltonian or \(G \in \mathcal{F}\), where \(K_{1,5}+e\) is a graph obtained by joining a pair of nonadjacent vertices in \(K_{s,s}\) and \(\mathcal{F} = \{G : K_{p,p-1} \subseteq G \subseteq K_{p} \vee (p+1)K_1, 2 \leq p \leq 3\}\), where \(\vee\) denotes the join operation of two graphs.

For a simple digraph \(D\) with \(n\) vertices, the energy of \(D\) is defined as \(E(D) = \sum_{i=1}^{n} |\Re(z_i)|\), where \(z_1, z_2, \ldots, z_n\) are the eigenvalues of \(D\). This paper first gives an improved lower bound on the spectral radius of \(D\), which is used to obtain some upper bounds for the energy \(E(D)\). These results improve and generalize some known results on upper bounds of the energy of digraphs.

A vertex \(v \in V(G)\) is said to be a self vertex switching of \(G\) if \(G\) is isomorphic to \(G^v\), where \(G^v\) is the graph obtained from \(G\) by deleting all edges of \(G\) incident to \(v\) and adding all edges incident to \(v\) which are not in \(G\). The set of all self vertex switchings of \(G\) is denoted by \({SS_1}(G)\) and its cardinality by \(ss_1(G)\). In [6], the number \(ss_1(G)\) is calculated for the graphs cycle, path, regular graph, wheel, Euler graph, complete graph, and complete bipartite graphs. In this paper, for a vertex \(v\) of a graph \(G\), the graph \(G^v\) is characterized for tree, star, and forest with a given number of components. Using this, we characterize trees and forests, each with a self vertex switching.

Permutation tableaux were introduced in the study of totally positive Grassmannian cells, and are connected with the steady state of asymmetric exclusion process, which is an important model from statistical mechanics. In this paper, we firstly establish a shape-preserving involution on the set of permutation tableaux of length \(n\), which directly shows that the number of permutation tableaux of length \(n\) with \(k\) essential 1’s equals the number of permutation tableaux of length \(n\) with \(n-k\) unrestricted rows. In addition, we introduce three combinatorial structures, called free permutation tableaux, restricted set partitions, and labeled Dyck paths. We discuss the properties of their internal structures and present the correspondence between the set of free permutation tableaux of length \(n\) and the set of restricted set partitions of \(\{1,2,\ldots,n\}\), and we also give a bijection between the set of restricted set partitions of \(\{1,2,\ldots,n\}\) and the set of labeled Dyck paths of length \(2n\). Finally, we make a generalization of the latter bijection.