We recall from [13] a shell graph of size \(n\), denoted \(C(n, n-3)\), is the graph obtained from the cycle \(C_n(v_1, v_2, \ldots, v_{n-1})\) by adding \(n-3\) consecutive chords incident at a common vertex, say \(v_0\). The vertex \(v_0\) of \(C(n, n-3)\) is called the apex of the shell \(C(n, n-3)\). The vertex \(v_1\) of \(C(n, n-3)\) is said to be at level 1.

A graph \(C(2n,n-2)\) is called an alternate shell, if \(C(2n,n-2)\) is obtained from the cycle \(C_{2n}(v_0,v_1, v_2, \ldots, v_{2n-1})\) by adding \(n-2\) chords between the vertex \(v_0\) and the vertices \(v_{2i+1}\), for \(1\leq i \leq n-2\). If the vertex \(v_i\) of \(C(2n,n-2)\) at level 1 is adjacent with \(v_0\), then \(v_1\) is said to be at level 1 with a chord, otherwise the vertex \(v_1\) is said to be at level 1 without a chord.

In 2009, Akelbek and Kirkland introduced a useful parameter called the scrambling index of a primitive digraph \(D\), which is the smallest positive integer \(k\) such that for every pair of vertices \(u\) and \(v\), there is a vertex \(w\) such that we can get to \(w\) from \(u\) and \(v\) in \(D\) by walks of length \(k\). In this paper, we study and obtain the scrambling indices of all primitive digraphs with exactly two cycles.

Given a tournament \(T = (V, A)\), a subset \(X\) of \(V\) is an interval of \(T\) provided that for every \(a, b \in X\) and \(x \in V – X\), \((a, x) \in A\) if and only if \((b, x) \in A\). For example, \(\emptyset\), \(\{x\}\) (\(x \in V\)), and \(V\) are intervals of \(T\), called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A critical tournament is an indecomposable tournament \(T\) of cardinality \(\geq 5\) such that for any vertex \(x\) of \(T\), the tournament \(T – x\) is decomposable. The critical tournaments are of odd cardinality and for all \(n \geq 2\) there are exactly three critical tournaments on \(2n + 1\) vertices denoted by \(T_{2n+1}\), \(U_{2n+1}\), and \(W_{2n+1}\). The tournaments \(T_5\), \(U_5\), and \(W_5\) are the unique indecomposable tournaments on 5 vertices. We say that a tournament \(T\) embeds into a tournament \(T’\) when \(T\) is isomorphic to a subtournament of \(T’\). A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and \(T_5\) embed into an indecomposable tournament \(T\), then \(W_5\) and \(U_5\) embed into \(T’\). To conclude, we prove the following: given an indecomposable tournament \(T\) with \(|V(T)| \geq 7\), \(T\) is critical if and only if only one of the tournaments \(T_7\), \(U_7\), or \(W_7\) embeds into \(T\).

Let \(\lambda K_{m,n}\) be a complete bipartite multigraph with two partite sets having \(m\) and \(n\) vertices, respectively. A \(K_{p,q}\)-factorization of \(\lambda K_{m,n}\) is a set of edge-disjoint \(K_{p,q}\)-factors of \(\lambda K_{m,n}\) which is a partition of the set of edges of \(\lambda K_{m,n}\). When \(\lambda = 1\), Martin, in paper [Complete bipartite factorisations by complete bipartite graphs, Discrete Math., \(167/168 (1997), 461-480]\), gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will give similar necessary conditions for \(\lambda K_{m,n}\) to have a \(K_{p,q}\)-factorization, and prove the necessary conditions are always sufficient in many cases.

In this paper, we determine upper and lower bounds for the number of independent sets in a bicyclic graph in terms of its order. This
gives an upper bound for the total number of independent sets in a connected graph which contains at least two cycles. In each case, we characterize the extremal graphs.

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 \(k\).