The Minimum Spectral Radius of Graphs with the Independence Number

In this paper, we investigate some properties of the Perron vector of connected graphs. These results are used to characterize that all extremal connected graphs with having the minimum (maximum) spectra radius among all connected graphs of order $n=k\alpha$ with the independence number $\alpha$, respectively.Comment: 28 pages, 3 figure

Independence in connected graphs

We prove that if $G=(V_G,E_G)$ is a finite, simple, and undirected graph with $\kappa$ components and independence number $\alpha(G)$, then there exist a positive integer $k\in \mathbb{N}$ and a function $f:V_G\to \mathbb{N}_0$ with non-negative integer values such that $f(u)\leq d_G(u)$ for $u\in V_G$, $\alpha(G)\geq k\geq \sum\limits_{u\in V_G}\frac{1}{d_G(u)+1-f(u)},$ and $\sum\limits_{u\in V_G}f(u)\geq 2(k-\kappa).$ This result is a best-possible improvement of a result due to Harant and Schiermeyer (On the independence number of a graph in terms of order and size, {\it Discrete Math.} {\bf 232} (2001), 131-138) and implies that $\frac{\alpha(G)}{n(G)}\geq \frac{2}{\left(d(G)+1+\frac{2}{n(G)}\right)+\sqrt{\left(d(G)+1+\frac{2}{n(G)}\right)2-8}}$ for connected graphs $G$ of order $n(G)$, average degree $d(G)$, and independence number $\alpha(G)$

On the largest real root of the independence polynomial of a unicyclic graph

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 \textit{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 $I(H,x)\ge I(G,x)\text{ for all }x\in [\xi(G),0].$ We find the maximum and minimum connected unicyclic and connected well-covered unicyclic graphs of a given order with respect to $\preceq$. This extends recent work by Oboudi 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 open questions posed by Oboudi and disprove a related conjecture due to Levit and Mandrescu