Open Journal of Discrete Applied Mathematics (ODAM)

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 \(A_{BS}(G)\).

We find the maximum and minimum connected unicyclic and connected well-covered unicyclic graphs of a given order with respect to \(\preceq\). This extends 2013 work by Csikv’ari where the maximum and minimum trees of a given order were determined and also answers an open question posed in the same work. Corollaries of our results give the graphs that minimize and maximize \(\xi(G)\) among all connected (well-covered) unicyclic graphs. We also answer more related open questions posed by Oboudi in 2018 and disprove a conjecture due to Levit and Mandrescu from 2008. The independence polynomial of a graph \(G\), denoted \(I(G,x)\), is the generating polynomial for the number of independent sets of each size. The roots of \(I(G,x)\) are called the independence roots of \(G\). It is known that for every graph \(G\), the independence root of smallest modulus, denoted \(\xi(G)\), is real. The relation \(\preceq\) on the set of all graphs is defined as follows, \(H\preceq G\) if and only if \(I(H,x)\ge I(G,x)\text{ for all }x\in [\xi(G),0].\)

In this paper, the fuzzy nonlinear partial differential equations of fractional order are considered. The generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) were applied to solve fuzzy nonlinear partial differential equations of fractional order. The above methods are investigated based on Taylor’s formula, and fuzzy Caputo’s fractional derivative. The proposed methods are also illustrated by some examples. The results reveal the methods are a highly effective scheme for obtaining the fuzzy fractional partial differential equations.