arXiv subjects

Ben Cameron

Publications and source records attributed to Ben Cameron.

24 records · Page 2Linked to original sources

arXiv2020

A note on purely imaginary independence roots

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each cardinality and its roots are called independence roots. We investigate here purely imaginary independence roots. We show that there are infinitely many connected graphs with purely imaginary independence roots and that every graph is a subgraph of such a graph. We also classify every rational purely imaginary number that is an independence root.

math.CO↗

arXiv2019

On the Mean Subtree Order of Graphs Under Edge Addition

For a graph $G$, the mean subtree order of $G$ is the average order of a subtree of $G$. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph $G$ and every pair of distinct vertices $u$ and $v$ of $G$, the addition of the edge between $u$ and $v$ increases the mean subtree order. In fact, we show that the addition of a single edge between a pair of nonadjacent vertices in a graph of order $n$ can decrease the mean subtree order by as much as $n/3$ asymptotically. We propose the weaker conjecture that for every connected graph $G$ which is not complete, there exists a pair of nonadjacent vertices $u$ and $v$, such that the addition of the edge between $u$ and $v$ increases the mean subtree order. We prove this conjecture in the special case that $G$ is a tree.

math.CO↗

arXiv2018

Maximum Modulus of Independence Roots of Graphs and Trees

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. We bound the maximum modulus, $\mbox{maxmod}(n)$, of an independence root over all graphs on $n$ vertices and the maximum modulus, $\mbox{maxmod}_{T}(n)$, of an independence root over all trees on $n$ vertices in terms of $n$. In particular, we show that $$\frac{\log_3(\mbox{maxmod}(n))}{n}=\frac{1}{3}+o(1)$$ and $$\frac{\log_2(\mbox{maxmod}_{T}(n))}{n}=\frac{1}{2}+o(1).$$

math.CO↗

arXiv2018

Independence Equivalence Classes of Paths and Cycles

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \textit{independence equivalent} if they have equivalent independence polynomials. We extend previous work by showing that independence equivalence class of every odd path has size 1, while the class can contain arbitrarily many graphs for even paths. We also prove that the independence equivalence class of every even cycle consists of two graphs when $n\ge 2$ except the independence equivalence class of $C_6$ which consists of three graphs. The odd case remains open, although, using irreducibility results from algebra, we were able show that for a prime $p \geq 5$ and $n\ge 1$ the independence equivalence class of $C_{p^n}$ consists of only two graphs.

math.CO↗

arXiv2018

On the Stability of Independence Polynomials

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions under which the roots lie in the left half-plane (all of the real roots of independence polynomial are negative and hence lie in this half-plane). We show stability for all independence polynomials of graphs with independence number at most three, but for larger independence number we show that the independence polynomials can have roots arbitrarily far to the right. We provide families of graphs whose independence polynomials are stable and ones that are not, utilizing various graph operations.

math.CO↗

arXiv2017

On the Unimodality of Independence Polynomials of Very Well-Covered Graphs

The independence polynomial $i(G,x)$ of a graph $G$ is the generating function of the numbers of independent sets of each size. A graph of order $n$ is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing). In this article we show that every graph is embeddable as an induced subgraph of a very well-covered graph whose independence polynomial is unimodal, by considering the location of the roots of such polynomials.

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