Vertices \(x\) and \(y\) are called paired in tournament \(T\) if there exists a vertex \(z\) in the vertex set of \(T\) such that either \(x\) and \(y\) beat \(z\) or \(z\) beats \(x\) and \(y\). Vertices \(x\) and \(y\) are said to be distinguished in \(T\) if there exists a vertex \(z\) in \(T\) such that either \(x\) beats \(z\) and \(z\) beats \(y\), or \(y\) beats \(z\) and \(z\) beats \(x\). Two vertices are strictly paired (distinguished) in \(T\) if all vertices of \(T\) pair (distinguish) the two vertices in question. The \(p/d\)-graph of a tournament \(T\) is a graph which depicts strictly paired or strictly distinguished pairs of vertices in \(T\). \(P/d\)-graphs are useful in obtaining the characterization of such graphs as domination and domination-compliance graphs of tournaments. We shall see that \(p/d\)-graphs of tournaments have an interestingly limited structure as we characterize them in this paper. In so doing, we find a method of constructing a tournament with a given \(p/d\)-graph using adjacency matrices of tournaments.

Let \(G\) be a simple connected graph. The spectral radius \(\rho(G)\) of \(G\) is the largest eigenvalue of its adjacency matrix. In this paper, we obtain two lower bounds of \(\rho(G)\) by two different methods, one of which is better than another in some conditions.

In this note we compute the chromatic polynomial of the Jahangir graph \(J_{2p}\) and we prove that it is chromatically unique for \(p=3\).

In this paper, we compute the PI and Szeged indices of some important classes of benzenoid graphs, which some of them are related to nanostructures. Some open questions are also included.

The detour \(d(i, j)\) between vertices \(i\) and \(j\) of a graph is the number of edges of the longest path connecting these vertices. The matrix whose \((i, j)\)-entry is the detour between vertices \(i\) and \(j\) is called the detour matrix. The half sum \(D\) of detours between all pairs of vertices (in a connected graph) is the detour index, i.e.,

\[D = (\frac{1}{2}) \sum\limits_j\sum\limits_i d(i,j)\]

In this paper, we computed the detour index of \(TUC_4C_8(S)\) nanotube.

We construct several new group divisible designs with block size five and with \(2, 3\), or \(6\) groups.

A list \((2,1)\)-labeling \(\mathcal{L}\) of graph \(G\) is an assignment list \(L(v)\) to each vertex \(v\) of \(G\) such that \(G\) has a \((2,1)\)-labeling \(f\) satisfying \(f(v) \in L(v)\) for all \(v\) of graph \(G\). If \(|L(v)| = k + 1\) for all \(v\) of \(G\), we say that \(G\) has a \(k\)-list \((2,1)\)-labeling. The minimum \(k\) taken over all \(k\)-list \((2,1)\)-labelings of \(G\), denoted \(\lambda_l(G)\), is called the list label-number of \(G\). In this paper, we study the upper bound of \(\lambda(G)\) of some planar graphs. It is proved that \(\lambda_l(G) \leq \Delta(G) + 6\) if \(G\) is an outerplanar graph or \(A\)-graph; and \(\lambda_l(G) \leq \Delta(G) + 9\) if \(G\) is an \(HA\)-graph or Halin graph.

In this paper, we give a necessary and sufficient condition for a \(3\)-regular graph to be cordial.

This paper deals with the interconnections between finite weakly superincreasing distributions, the Fibonacci sequence, and Hessenberg matrices. A frequency distribution, to be called the Fibonacci distribution, is introduced that expresses the core of the connections among these three concepts. Using a Hessenberg representation of finite weakly superincreasing distributions, it is shown that, among all such \(n\)-string frequency distributions, the Fibonacci distribution achieves the maximum expected codeword length.

We present some applications of wall colouring to scheduling issues. In particular, we show that the chromatic number of walls has a very clear meaning when related to certain real-life situations.