Note on the smallest root of the independence polynomial

One can define the independence polynomial of a graph G as follows. Let i(k)(G) denote the number of independent sets of size k of G, where i(0)(G) = 1. Then the independence polynomial of G is I(G,x) = Sigma(n)(k=0)(-1)(k)i(k)(G)x(k). In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real

Clique polynomials of 22-connected K5K_{5}-free chordal graphs

In this paper, we give a generalization of the author's previous result about real rootedness of clique polynomials of connected $K_{4}$-free chordal graphs to the class of $2$-connected $K_{5}$-free chordal graphs. The main idea is based on the graph-theoretical interpretation of the second derivative of clique polynomials. Finally, we conclude the paper with several interesting open questions and conjectures

The zero forcing polynomial of a graph

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph $G$ of order $n$ as the polynomial $\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i$, where $z(G;i)$ is the number of zero forcing sets of $G$ of size $i$. We characterize the extremal coefficients of $\mathcal{Z}(G;x)$, derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of $\mathcal{Z}(G;x)$, including multiplicativity, unimodality, and uniqueness.Comment: 23 page

Alliance polynomial and hyperbolicity in regular graphs

One of the open problems in graph theory is the characterization of any graph by a polynomial. Research in this area has been largely driven by the advantages offered by the use of computers which make working with graphs: it is simpler to represent a graph by a polynomial (a vector) that by the adjacency matrix (a matrix). We introduce the alliance polynomial of a graph. The alliance polynomial of a graph G with order n and maximum degree δ_1 is the polynomial A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) 〗, where A{_k}(G) is the number of exact defensive k-alliances in G. Also, we develop and implement an algorithm that computes in an efficient way the alliance polynomial. We obtain some properties of A(G; x) and its coefficients for: • Path, cycle, complete and star graphs. In particular, we prove that they are characterized by their alliance polynomials. • Cubic graphs (graphs with all of their vertices of degree 3), since they are a very interesting class of graphs with many applications. We prove that they verify unimodality. Also, we compute the alliance polynomial for cubic graphs of small order, which satisfy uniqueness. • Regular graphs (graphs with the same degree for all vertices). In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of connected ∆-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not connected ∆-regular. If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in the 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 >= 0 : X is δ-hyperbolic }. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. We obtain information about the hyperbolicity constant of cubic graphs. These graphs are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is the hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants. Besides, we obtain bounds for the hyperbolicity constant of the complement graph of a cubic graph; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to K_4 or to K_3,3, the inequalities 5k/4 <= δ (G ̅) <=3k/2 hold, if k is the length of every edge in G. --------------------Uno de los problemas abiertos en la teoría de grafos es la caracterización de cualquier grafo por un polinomio. La investigación en este área ha sido impulsada en gran parte por las ventajas que ofrece el uso de las computadoras que hacen que trabajar con grafos sea más simple. En esta Tesis introducimos el polinomio de alianza de un grafo. El polinomio de alianza de un grafo G con orden n y grado m´aximo δ_1 es el polinomio A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) , donde A{_k}(G) es el n´umero de k alianzas defensivas exactas en G. También desarrollamos e implementamos un algoritmo que calcula de manera eficiente el polinomio de alianza. En este trabajo obtenemos algunas propiedades de A(G; x) y sus coeficientes para: • Grafos caminos, ciclos, completos y estrellas. En particular, hemos demostrado que se caracterizan mediante sus polinomios de alianza. • Grafos cúbicos (grafos con todos sus vértices de grado 3), ya que son una clase muy interesante de grafos con muchas aplicaciones. Hemos demostrado que sus polinomios de alianza verifican unimodalidad. Además, calculamos el polinomio de alianza para grafos cúbicos de orden pequeño, los cuales satisfacen unicidad. • Grafos regulares (grafos con todos sus vértices de igual grado). En particular, se caracteriza el grado de los grafos regulares por el n´umero de coeficientes distintos de cero de su polinomio de alianza. Además, se demuestra que la familia de polinomios de alianza de grafos conexos _-regulares con grado pequeño es muy especial, ya que no contiene polinomios de alianza de grafos conexos que no sean _-regulares. Si X es un espacio métrico geodésico y x1, x2, x3 ∈ X, un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] and [x3x1] de X. El espacio X es δ-hiperbólico (en el sentido de Gromov) si todo lado de todo triángulo geodésico T de X está contenido en la δ-vecindad de la unóon de los otros dos lados. Se denota por δ(X) la constante de hiperbolicidad óptima de X, es decir, δ(X) := inf{δ > 0 : X es δ-hiperbólico }. El estudio de los grafos hiperbólicos es un tema interesante dado que la hiperbolicidad de un espacio métrico geodésico es equivalente a la hiperbolicidad de un grafo más sencillo asociado al espacio. Hemos obtenido información acerca de la constante de hiperbolicidad de los grafos cúbicos; dichos grafos son muy importantes en el estudio de la hiperbolicidad, ya que para cualquier grafo G con grado máximo acotado existe un grafo cúbico G∗ tal que G es hiperbólico si y sólo si G∗ es hiperbólico. En esta memoria conseguimos caracterizar los grafos cúbicos con constante de hiperbolicidad pequeña. Además, se obtienen cotas para la constante de hiperbolicidad del grafo complemento de un grafo cúbico; nuestro principal resultado dice que para cualquier grafo cúbico finito G no isomorfo a K4 o K3,3, se cumple la relación 5k/4 <= δ (G ̅) <=3k/2, donde k es la longitud de todas las aristas en G.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Domingo de Guzmán Pestana Galván; Secretario: Eva Tourís Lojo; Vocal: Sergio Bermudo Navarret