Let \(G\) be a connected graph of order \(n\). Denote \(p_u(G)\) the order of a longest path starting at vertex \(u\) in \(G\). In this paper, we prove that if \(G\) has more than \(t\binom{k}{2} + \binom{p+1}{2} + (n-k-1)\) edges, where \(k \geq 2\), \(n = t(k-1) + p + 1\), \(t \geq 0\) and \(0 \leq p \leq k-1\), then \(p_u(G) > k\) for each vertex \(u\) in \(G\). By this result, we give an alternative proof of a result obtained by P. Wang et al. that if \(G\) is a 2-connected graph on \(n\) vertices and with more than \(t\binom{k-2}{2} + \binom{p}{2} + (2n – 3)\) edges, where \(k \geq 3\), \(n-2 = t(k-2) + p\), \(t \geq 0\) and \(0 \leq p \leq k-2\), then each edge of \(G\) lies on a cycle of order more than \(k\).

In this paper, we give some identities involving the harmonic numbers and the inverses of binomial coefficients.

In this paper, a new efficient computational algorithm is presented for solving cyclic heptadiagonal linear systems based on using the heptadiagonal linear solver and Sherman–Morrison–Woodbury formula. The implementation of the algorithm using computer algebra systems (CAS) such as MAPLE and MATLAB is straightforward. Two numerical examples are presented for illustration.

Let \(G\) be a graph, and let \(a, b\), and \(k\) be nonnegative integers with \(0 \leq a \leq b\). A graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\), the remaining graph of \(G\) has an \([a, b]\)-factor. In this paper, we prove that if \(\delta(G) \geq a + k\) and \(\alpha(G) \leq \frac{4b(\delta(G)-a+1-1)}{(a+1)^2}\), then \(G\) is an \((a, b, k)\)-critical graph. Furthermore, it is shown that the result in this paper is best possible in some sense.

A characterization of \(B\)-H-unretractive bipartite graphs is given. Based on this, it is proved that there is no bipartite graph with endotype \(1 \pmod{4}\).

In a graph \(G = (V, E)\), an independent set is a subset \(I\) of \(V(G)\) such that no two vertices in \(I\) are adjacent. A maximum independent set is an independent set of maximum size. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we study the problem of determining the largest and the second largest numbers of maximum independent sets among all quasi-tree graphs and quasi-forest graphs. Extremal graphs achieving these values are also given.

The notions of sum labelling and sum number of graphs were introduced by F. Harary [1] in 1990. A mapping \(f\) is called a sum labelling of a graph \(G(V, E)\) if it is an injection from \(V\) to a set of positive integers such that \(uv \in E\) if and only if there exists a vertex \(w \in V\) such that \(f(w) = f(x) + f(y)\). In this case, \(w\) is called a working vertex. If \(f\) is a sum labelling of \(G\) with \(r\) isolated vertices, for some nonnegative integer \(r\), and \(G\) contains no working vertex, \(f\) is defined as an exclusive sum labelling of the graph \(G\) by M. Miller et al. in paper [2]. The least possible number \(r\) of such isolated vertices is called the exclusive sum number of \(G\), denoted by \(\epsilon(G)\). If \(\epsilon(G) = \Delta(G)\), the labelling is called \(\Delta\)-optimum exclusive sum labelling and the graph is said to be \(\Delta\)-optimum summable, where \(\Delta = \Delta(G)\) denotes the maximum degree of vertices in \(G\). By using the notion of \(\Delta\)-optimum forbidden subgraph of a graph, the exclusive sum numbers of crown \(C_n \odot K_1\) and \((C_n \odot K_1)\) are given in this paper. Some \(\Delta\)-optimum forbidden subgraphs of trees are studied, and we prove that for any integer \(\Delta \geq 3\), there exist trees not \(\Delta\)-optimum summable. A nontrivial upper bound of the exclusive sum numbers of trees is also given in this paper.

In this paper we obtain the Fibonacci length of amalgamated free products having as factors dihedral groups.

In [11], Zhu, Li, and Deng introduced the definition of implicit degree of a vertex \(v\), denoted by \(\text{id}(v)\). In this paper, we consider implicit degrees and the hamiltonicity of graphs and obtain that:
If \(G\) is a \(2\)-connected graph of order \(n\) such that \(\text{id}(u) + \text{id}(v) \geq n – 1\) for each pair of vertices \(u\) and \(v\) at distance \(2\), then \(G\) is hamiltonian, with some exceptions.

Let \(C_k\) denote a cycle of length \(k\) and let \(S_k\) denote a star with \(k\) edges. For graphs \(F\), \(G\), and \(H\), a \((G, H)\)-multidecomposition of \(F\) is a partition of the edge set of \(F\) into copies of \(G\) and copies of \(H\) with at least one copy of \(G\) and at least one copy of \(H\). In this paper, necessary and sufficient conditions for the existence of the \((C_k, S_k)\)-multidecomposition of a complete bipartite graph are given.