For a vertex \(v\) of a graph \(G\), Zhu, Li, and Deng introduced the concept of implicit degree \(id(v)\), according to the degrees of its neighbors and the vertices at distance \(2\) with \(v\) in \(G\). For \(S \subset V(G)\), let \(i\Delta_2(S)\) denote the maximum value of the implicit degree sum of two vertices of \(S\). In this paper, we will prove the following result: Let \(G\) be a \(2\)-connected graph on \(n \geq 3\) vertices. If \(i\Delta_2(S) \geq d\) for each independent set \(S\) of order \(\kappa(G) + 1\), then \(G\) has a cycle of length at least \(\min\{d, n\}\). This result generalizes one result of Yamashita [T. Yamashita, On degree sum conditions for long cycles and cycles through specified vertices, Discrete Math., \(308 (2008) 6584-6587]\).

For a given graph \(G = (V, E)\), by \(f(v)\), we denote the sum of the color on the vertex \(v\) and the colors on the edges incident with \(v\). A proper \(k\)-total coloring \(\phi\) of a graph \(G\) is called a neighbor sum distinguishing \(k\)-total coloring if \(f(u) \neq f(v)\) for each edge \(uv \in E(G)\). The smallest number \(k\) in such a coloring of \(G\) is the neighbor sum distinguishing total chromatic number, denoted by \(\chi”_{\sum}(G)\). The maximum average degree of \(G\) is the maximum of the average degree of its non-empty subgraphs, which is denoted by \(\mathrm{mad}(G)\). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if \(G\) is a graph with \(\Delta(G) \geq 6\) and \(\mathrm{mad}(G) < \frac{18}{5}\), then \(\chi''_{\sum}(G) \leq \Delta(G) + 2\). This bound is sharp.

A two-character set is a set of points of a finite projective space that has two intersection numbers with respect to hyperplanes. Two-character sets are related to strongly regular graphs and two-weight codes. In the literature, there are plenty of constructions for (non-trivial) two-character sets by considering suitable subsets of quadrics and Hermitian varieties. Such constructions exist for the quadrics \(Q^{+}(2n-1,4) \subseteq PG(2n-1,q)\), \(Q^{-}(2n+1,4) \subseteq PG(2n+1,q)\) and the Hermitian varieties \(H(2n-1,q^{2}) \subseteq PG(2n-1,q^{2})\), \(H(2n,q^{2}) \subseteq PG(2n,q^{2})\). In this note, we show that every two-character set of \(PG(2n,q)\) that is contained in a given nonsingular parabolic quadric \(Q(2n,q) \subseteq PG(2n,q)\) is a subspace of \(PG(2n,q)\). This offers some explanation for the absence of the parabolic quadrics in the above-mentioned constructions.

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.