Open Journal of Discrete Applied Mathematics (ODAM)

The atom-bond connectivity (ABC) index of a graph \(G=(V,E)\) is defined as \(ABC(G)=\sum_{v _{i}v_{j} \in E}\sqrt{(d_{i}+d_{j}-2)/(d_{i}d_{j})}\), where \(d_{i}\) denotes the degree of vertex \(v_{i}\) of \(G\). Due to its interesting applications in chemistry, this molecular structure descriptor has become one of the most actively studied vertex-degree-based graph invariants. Many efforts were made towards the elementary problem of characterizing tree(s) with minimal ABC index, which remains open and was coined as the ABC index conundrum”. Up to date, quite a few significant results have been obtained. In the course of research computer search plays a non-negligible role. In the present paper we review the state of the art of the problem. In addition we intend to demonstrate that, repeating the procedure “searching – conjecturing – proving” can be an applicable paradigm to cope with elusive problems of extremal graph characterization.

Let \(G\) be a simple, finite and connected graph. A graph is said to be decomposed into subgraphs \(H_1\) and \(H_2\) which is denoted by \(G= H_1 \oplus H_2\), if \(G\) is the edge disjoint union of \(H_1\) and \(H_2\). Assume that \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\) and if each (H_i\), \(1 \leq i \leq k\), is a path or cycle in \(G\), then the collection of edge-disjoint subgraphs of \(G\) denoted by \(\psi\) is called a path decomposition of \(G\). If each \(H_i\) is a path in \(G\) then \(\psi\) is called an acyclic path decomposition of \(G\). The minimum cardinality of a path decomposition of \(G\), denoted by \(\pi (G)\), is called the path decomposition number and the minimum cardinality of an acyclic path decomposition of \(G\), denoted by \(\pi_a(G)\), is called the acyclic path decomposition number of \(G\). In this paper, we determine path decomposition number for a number of graphs in particular, the Cartesian product of graphs. We also provided bounds for \(\pi(G)\) and \(\pi_a(G)\) for these graphs.

For an undirected graph \(G\), a zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum \(k\)-flow if the absolute values of edges are less than \(k\). We define the zero-sum flow number of \(G\) as the least integer \(k\) for which \(G\) admitting a zero sum \(k\)-flow. In this paper we gave complete zero-sum flow and zero sum number for octagonal grid, generalized prism and book graph.

Let \(V(G) = \{v_1, v_2, \ldots, v_n\}\) be the vertex set of \(G\) and let \(d_{G}(v_i)\) be the degree of a vertex \(v_i\) in \(G\). The degree subtraction adjacency matrix of \(G\) is a square matrix \(DSA(G)=[d_{ij}]\), in which \(d_{ij}=d_{G}(v_i)-d_{G}(v_j)\), if \(v_i\) is adjacent to \(v_j\) and \(d_{ij}=0\), otherwise. In this paper we express the eigenvalues of the degree subtraction adjacency matrix of subdivision graph, semitotal point graph, semitotal line graph and total graph of a regular graph in terms of the adjacency eigenvalues of \(G\). Further we obtain the degree subtraction adjacency energy of these graphs.

The zeroth-order general Randić index of a simple connected graph G is defined as \(R_{\alpha}^{0}(G)=\sum_{u\in V(G)} \big(d(u)\big)^{\alpha}\), where \(d(u)\) is the degree of \(u\) and \(\alpha\not\in \{0,1\}\) is a real number. A \(k\)-polygonal cactus is a connected graph in which every edge lies in exactly one cycle of length \(k\). In this paper, we present the extremal \(k\)-polygonal cactus with \(n\) cycles for \(k\geq3\) with respect to the zeroth-order general Randić index.