Open Journal of Discrete Applied Mathematics (ODAM)

In a simple connected graph \(G\), eccentricity of a vertex is one of the first, distance-based invariants. The eccentricity of a vertex \(v\) in a connected graph \(G\) is the maximum distance of the vertex \(v\) to any other vertex \(u\). The total eccentricity of the graph \(G\) is the sum of the all vertex eccentricities. A graph \(G\) is called an apex tree if it has a vertex \(x\) such that \(G-x\) is a tree. In this work we have found the graph having extremal total eccentricity of \(k\)-apex trees.

In this paper, we obtained some new properties of Zagreb indices. We mainly give explicit formulas to the second Zagreb index of semitotal-line graph (or middle graph), semitotal-point graph and total transformation graphs \(G^{xyz}.\)

A graceful difference labeling (gdl for short) of a directed graph \(G\) with vertex set \(V\) is a bijection \(f:V\rightarrow\{1,\ldots,\vert V\vert\}\) such that, when each arc $uv$ is assigned the difference label \(f(v)-f(u)\), the resulting arc labels are distinct. We conjecture that all disjoint unions of circuits have a gdl, except in two particular cases. We prove partial results which support this conjecture.

Given a set of locations or cities and the cost of travel between each location, the task is to find the optimal tour that will visit each locations exactly once and return to the starting location. We solved a routing problem with focus on Traveling Salesman Problem using two algorithms. The task of choosing the algorithm that gives optimal result is difficult to accomplish in practice. However, most of the traditional methods are computationally bulky and with the rise of machine learning algorithms, which gives a near optimal solution. This paper studied two methods: branch-and-cut and machine learning methods. In the machine learning method, we used neural networks and reinforcement learning with 2-opt to train a recurrent network that predict a distribution of different location permutations using the negative tour-length as the reward signal and policy gradient to optimize the parameters of recurrent network. The improved machine learning with 2-opt give near-optimal results on 2D Euclidean with upto 200 nodes.

The Wiener index \(W(G)\) of a graph \(G\) is defined as the sum of distances between its vertices. A tree \(T\) generates \(r\)-uniform hypergraph \(H_{r,k}(T)\) by the following way: hyperedges of cardinality \(r\) correspond to edges of the tree and adjacent hyperedges have \(k\) vertices in common. A relation between quantities \(W(T)\) and \(W(H_{r,k}(T))\) is established.

The aim of this paper is to calculate the multiplicative topological indices of Zigzag polyhex nanotubes, Armchair polyhex nanotubes, Carbon nanocone networks, two dimensional Silicate network, Chain silicate network, six dimensional Hexagonal network, five dimensional Oxide network and four dimensional Honeycomb network.

Let \(G\) be a simple connected graph with \(n\) vertices, \(m\) edges, and a sequence of vertex degrees \(\Delta=d_1\geq d_2\geq\cdots\geq d_n=\delta >0\). Denote by \(\mu_1\geq \mu_2\geq\cdots\geq \mu_{n-1}>\mu_n=0\) the Laplacian eigenvalues of \(G\). The Kirchhoff index of \(G\) is defined as \(Kf(G)=n\sum_{i=1}^{n-1} \frac{1}{\mu_i}\). A couple of new lower bounds for \(Kf(G)\) that depend on \(n\), \(m\), \(\Delta\) and some other graph invariants are obtained.

In this note, we first show that the general Zagreb index can be obtained from the \(M-\)polynomial of a graph by giving a suitable operator. Next, we obtain \(M-\)polynomial of some cactus chains. Furthermore, we derive some degree based topological indices of cactus chains from their \(M-\)polynomial.

Let \(G= G(V,E)\) be a \((p,q)\)-graph. A bijection \(f: E\to\{1,2,3,\ldots,q \}\) is called an edge-prime labeling if for each edge \(uv\) in \(E\), we have \(GCD(f^+(u),f^+(v))=1\) where \(f^+(u) = \sum_{uw\in E} f(uw)\). A graph that admits an edge-prime labeling is called an edge-prime graph. In this paper we obtained some sufficient conditions for graphs with regular component(s) to admit or not admit an edge-prime labeling. Consequently, we proved that if \(G\) is a cubic graph with every component is of order \(4, 6\) or \(8\), then \(G\) is edge-prime if and only if \(G\not\cong K_4\) or \(nK(3,3)\), \(n\equiv2,3\pmod{4}\). We conjectured that a connected cubic graph \(G\) is not edge-prime if and only if \(G\cong K_4\).

In this article, we study some properties of the solutions of the following difference equation: \(x_{n+1}=a x_{n}+\dfrac{b x_{n} x_{n-4}}{cx_{n-3}+dx_{n-4}},\quad n=0,1,…\) where the initial conditions \(x_{-4},x_{-3}, x_{-2}, x_{-1}, x_0\) are arbitrary positive real numbers and \(a, b, c, d\) are positive constants. Also, we give specific form of the solutions of four special cases of this equation.