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.

Let \(G\) be a connected graph. For \(S \subseteq V(G)\), the geodetic closure \(I_G[S]\) of \(S\) is the set of all vertices on geodesics (shortest paths) between two vertices of \(S\). We select vertices of \(G\) sequentially as follows: Select a vertex \(v_1\) and let \(S_1 = \{v_1\}\). Select a vertex \(v_2 \neq v_1\) and let \(S_2 = \{v_1, v_2\}\). Then successively select vertex \(v_i \notin I_G[S_{i-1}]\) and let \(S_i = \{v_1, v_2, \ldots, v_i\}\). We define the closed geodetic number (resp. upper closed geodetic number) of \(G\), denoted \(cgn(G)\) (resp. \(ucgn(G)\)), to be the smallest (resp. largest) \(k\) whose selection of \(v_1, v_2, \ldots, v_k\) in the given manner yields \(I_G[S_k] = V(G)\). In this paper, we show that for every pair \(a, b\) of positive integers with \(2 \leq a \leq b\), there always exists a connected graph \(G\) such that \(cgn(G) = a\) and \(ucgn(G) = b\), and if \(a < b\), the minimum order of such graph \(G\) is \(b\). We characterize those connected graphs \(G\) with the property: If \(cgn(G) < k < ucgn(G) = 6\), then there is a selection of vertices \(v_1, v_2, \ldots, v_k\) as in the above manner such that \(I_G[S_k] = V(G)\). We also determine the closed and upper closed geodetic numbers of some special graphs and the joins of connected graphs.

Let \(G\) be a graph with \(n\) vertices and suppose that for each vertex \(v\) in \(G\), there exists a list of \(k\) colors, \(L(v)\), such that there is a unique proper coloring for \(G\) from this collection of lists, then \(G\) is called a uniquely \(k\)-list colorable graph. We say that a graph \(G\) has the property \(M(k)\) if and only if it is not uniquely \(k\)-list colorable. M. Ghebleh and E. S. Mahmoodian characterized uniquely \(3\)-list colorable complete multipartite graphs except for the graphs \(K_{1*4,5}\), \(K_{1*5,4}, K_{1*4,4}\), \(K_{2,3,4}\), and \(K_{2,2,r}\), \(4 \leq r \leq 8\). In this paper, we prove that the graphs \(K_{1*4,5}\), \(K_{1*5,4}\), \(K_{1*4,4}\), and \(K_{2,3,4}\) have the property \(M(3)\).

Let \(G\) be a simple graph and \(f: V(G) \mapsto \{1, 3, 5, \ldots\}\) an odd integer valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called a \((1, f)\)-odd factor if \(d_F(v) \in \{1, 3, \ldots, f(v)\}\) for all \(v \in V(G)\), where \(d_F(v)\) is the degree of \(v\) in \(F\). For an odd integer \(k\), if \(f(v) = k\) for all \(v\), then a \((1, f)\)-odd factor is called a \([1, k]\)-odd factor. In this paper, the structure and properties of a graph with a unique \((1, f)\)-odd factor is investigated, and the maximum number of edges in a graph of the given order which has a unique \([1, k]\)-odd factor is determined.