Open Journal of Discrete Applied Mathematics (ODAM)

A \(2\)-noncrossing tree is a rooted tree drawn in the plane with its vertices (colored black or white) on the boundary of a circle such that the edges are line segments that do not intersect inside the circle and there is no black-black ascent in any path from the root. A rooted tree is said to be increasing if the labels of the vertices are increasing as one moves away from the root. In this paper, we use generating functions and bijections to enumerate \(2\)-noncrossing increasing trees by the number of blacks vertices and by root degree. Bijections with noncrossing trees, ternary trees, 2-plane trees, certain Dyck paths, and certain restricted lattice paths are established.

This paper introduces the new notion of total chromatic vertex stress of a graph. Results for certain tree families and other \(2\)-colorable graphs are presented. The notions of chromatically-stress stability and chromatically-stress regularity are also introduced. New research avenues are also proposed.

Coloring the arcs of biregular graphs was introduced with possible applications to industrial chemistry, molecular biology, cellular neuroscience, etc. Here, we deal with arc coloring in some non-bipartite graphs. In fact, for \(1<k\in\mathbb{Z}\), we find that the odd graph \(O_k\) has an arc factorization with colors \(0,1,\ldots,k\) such that the sum of colors of the two arcs of each edge equals \(k\). This is applied to analyzing the influence of such arc factorizations in recently constructed uniform 2-factors in \(O_k\) and in Hamilton cycles in \(O_k\) as well as in its double covering graph known as the middle-levels graph \(M_k\).

Let \(G\) be a graph with \(n\) vertices. The second Zagreb energy of graph \(G\) is defined as the sum of the absolute values of the eigenvalues of the second Zagreb matrix of graph \(G\). In this paper, we derive the relation between the second Zagreb matrix and the adjacency matrix of graph \(G\) and derive the new upper bound for the second Zagreb energy in the context of trace. We also derive the second Zagreb energy of \(m-\)splitting graph and \(m-\)shadow graph of a graph.

The Sombor index (\(SO\)) and the modified Sombor index (\(^mSO\)) are two closely related vertex-degree-based graph invariants. Both were introduced in the 2020s, and have already found a variety of chemical, physicochemical, and network-theoretical applications. In this paper, we examine the product \(SO \cdot {^mSO}\) and determine its main properties. It is found that the structure-dependence of this product is fully different from that of either \(SO\) or \(^mSO\). Lower and upper bounds for \(SO \cdot {^mSO}\) are established and the extremal graphs are characterized. For connected graphs, the minimum value of the product \(SO \cdot {^mSO}\) is the square of the number of edges. In the case of trees, the maximum value pertains to a special type of eclipsed sun graph, trees with a single branching point.

Using coalescence and cones, this study defines three types of graphs formed by amalgamating vertices of disjoint unions of complete graphs. The three types include the cone over a disjoint union of two complete graphs (C1), the cone over a disjoint union of \(k\) complete graphs (C2), and the \(l\) cone over a disjoint union of two complete graphs (C3). Coalescence of complete graphs (C1, C3) and the \(l\) cone (C3) are determined by their Laplacian spectra, a novel finding. Their Laplacian spectra reveal the size of the vertex cutset. Applications include the analysis of corporate networks, where individuals form coalescence of complete graphs through joint membership of two or more company boards.

A famous conjecture of Ryser states that any \(r\)-partite set system has transversal number at most \(r-1\) times their matching number. This conjecture is only known to be true for \(r\leq3\) in general, for \(r\leq5\) if the set system is intersecting, and for \(r\leq9\) if the set system is intersecting and linear. In this note, we deal with Ryser’s conjecture for intersecting \(r\)-partite linear systems: if \(\tau\) is the transversal number for an intersecting \(r\)-partite linear system, then \(\tau\leq r-1\). If this conjecture is true, this is known to be sharp for \(r\) for which there exists a projective plane of order \(r-1\). There has also been considerable effort to find intersecting \(r\)-partite set systems whose transversal number is \(r-1\). In this note, we prove that if \(r\geq2\) is an even integer, then \(f_l(r)\geq3r-5\), where \(f_l(r)\) is the minimum number of lines of an intersecting \(r\)-partite linear system whose transversal number is \(r-1\). Aharoni \emph{et al.,} [R. Aharoni, J. Barát and I.M. Wanless, \emph{Multipartite hypergraphs achieving equality in Ryser’s conjecture}, Graphs Combin. {\bf 32}, 1–15 (2016)] gave an asymptotic lower bound: \(f_l(r)\geq3\).\(052r+O(1)\) as \(r\to\infty\). For some small values of \(r\) (\(r\geq2\) an even integer), our lower bound is better. Also, we prove that any \(r\)-partite linear system satisfies \(\tau\leq r-1\) if \(\nu_2\leq r\) for all \(r\geq3\) odd integer and \(\nu_2\leq r-2\) for all \(r\geq4\) even integer, where \(\nu_2\) is the maximum cardinality of a subset of lines \(R\subseteq\mathcal{L}\) such that any three elements chosen in \(R\) do not have a common point.

The vertex-degree based (VDB) topological index (or graphical function-index) \(TI_{f}(G)\) of \(G\) with edge-weight function \(f(x,y)\) was defined as \(TI_{f}(G)=\sum\limits_{uv\in E(G)}f(d(u),d(v))\), where \(d(u)\) is the degree of vertex \(u\) in \(G\). In this paper, we use a unified way to determine the extremal values of VDB indices of connected \((n,m)\)-graphs. When \(f(x,y)\) satisfies some special properties, we determine the connected \((n,m)\)-graphs with maximum (or minimum) \(TI_{f}(G)\) is the almost regular graphs. Our results generalize the results of paper [Aashtab, A., Akbari, S., Madadinia, S., Noei, M., \& Salehi, F. (2022) On the graphs with minimum Sombor index. MATCH Commun. Math. Comput. Chem., {88}, 553-559].

For all positive even integers \(n\), graphs of order \(n\) with degree sequence \(S_{n}:1,2,\dots,n/2,n/2,n/2+1,n/2+2,\dots,n-1\) naturally arose in the study of a labeling problem in [1].This fact motivated the authors of the aforementioned paper to study these sequences and as a result of this study they proved that there is a unique graph of order \(n\) realizing \(S_{n}\) for every even integer \(n\). The main goal of this paper is to generalize this result.

The Sombor index has gained lot of attention in the recent days for its mathematical properties and chemical applicabilities. Here, we initiated the novel block number version of the classical Sombor index and its matrix representation of a graph. The Block Sombor index \(BS(G)\) is defined as the sum total of square root of the sum of squares of block numbers of adjacent vertices, where the block number of a vertex is the number of blocks to which that vertex belongs to. The main purpose of this paper is to obtain some bounds and characterizations of \(BS(G)\) and its Block Sombor energy \(E_{BS}\). Also, we estimate some properties of spectral radius of Block Sombor matrix \(A_{BS}(G)\).