Walter Carballosa — Page 2

arXiv2014

Computing the alliance polynomial of a graph

The alliance polynomial of a graph $Γ$ with order $n$ and maximum degree $δ_1$ is the polynomial $A(Γ; x) = \sum_{k=-δ_1}^{δ_1} A_{k}(Γ) \, x^{n+k}$, where $A_{k}(Γ)$ is the number of exact defensive $k$-alliances in $Γ$. We provide an algorithm for computing the alliance polynomial. Furthermore, we obtain some properties of $A(Γ; x)$ and its coefficients. In particular, we prove that the path, cycle, complete and star graphs are characterized by their alliance polynomials. We also show that the alliance polynomial characterizes many graphs that are not distinguished by other usual polynomials of graphs.

math.CO↗

arXiv2014

New inequalities on the hyperbolicity constant of line graphs

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $δ$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. We denote by $δ(X)$ the sharp hyperbolicity constant of $X$, i.e. $δ(X):=\inf\{δ\ge 0: \, X \, \text{ is $δ$-hyperbolic}\,\}\,. $ The main result of this paper is the inequality $δ(G) \le δ(\mathcal L(G))$ for the line graph $\mathcal L(G)$ of every graph $G$. We prove also the upper bound $δ(\mathcal L(G)) \le 5 δ(G)+ 3 l_{max}$, where $l_{max}$ is the supremum of the lengths of the edges of $G$. Furthermore, if every edge of $G$ has length $k$, we obtain $δ(G) \le δ(\mathcal L(G)) \le 5 δ(G)+ 5k/2$.

math.CO↗