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 \(I(H,x)\ge I(G,x)\text{ for all }x\in [\xi(G),0].\)
This paper presents several fixed point theorems for intuitionistic generalized fuzzy metric spaces with an implicit relation. Specifically, we utilize compatible maps of type \((\beta)\) in intuitionistic generalized fuzzy metric spaces to derive our fixed point theorems. Our results not only extend but also generalize some fixed point theorems that were previously established in complete fuzzy metric spaces. This is achieved by introducing a novel technique, which enhances the applicability and scope of the existing fixed point theorems.
