Defensive alliance polynomial

We introduce a new bivariate polynomial which we call the defensive alliance polynomial and denote it by da(G; x, y). It is a generalization of the alliance polynomial [Carballosa et al., 2014] and the strong alliance polynomial [Carballosa et al., 2016]. We show the relation between da(G; x, y) and the alliance, the strong alliance and the induced connected subgraph [Tittmann et al., 2011] polynomials. Then, we investigate information encoded in da(G; x, y) about G. We discuss the defensive alliance polynomial for the path graphs, the cycle graphs, the star graphs, the double star graphs, the complete graphs, the complete bipartite graphs, the regular graphs, the wheel graphs, the open wheel graphs, the friendship graphs, the triangular book graphs and the quadrilateral book graphs. Also, we prove that the above classes of graphs are characterized by its defensive alliance polynomial. A relation between induced subgraphs with order three and both subgraphs with order three and size three and two respectively, is proved to characterize the complete bipartite graphs. Finally, we present the defensive alliance polynomial of the graph formed by attaching a vertex to a complete graph. We show two pairs of graphs which are not characterized by the alliance polynomial but characterized by the defensive alliance polynomial

An Abstraction of Whitney's Broken Circuit Theorem

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the M\"obius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical M\"obius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).Comment: 18 page

Polynomial graph invariants from homomorphism numbers

We give a new method of generating strongly polynomial sequences of graphs, i.e., sequences (Hk) indexed by a tuple k = (k1, . . . , kh) of positive integers, with the property that, for each fixed graph G, there is a multivariate polynomial p(G; x1, . . . , xh) such that the number of homomorphisms from G to Hk is given by the evaluation p(G; k1, . . . , kh). A classical example is the sequence of complete graphs (Kk), for which p(G; x) is the chromatic polynomial of G. Our construction is based on tree model representations of graphs. It produces a large family of graph polynomials which includes the Tutte polynomial, the Averbouch–Godlin–Makowsky polynomial, and the Tittmann–Averbouch–Makowsky polynomial. We also introduce a new graph parameter, the branching core size of a simple graph, derived from its representation under a particular tree model, and related to how many involutive automorphisms it has. We prove that a countable family of graphs of bounded branching core size is always contained in the union of a finite number of strongly polynomial sequences.Ministerio de Economía y Competitividad MTM2014-60127-

The enumeration of vertex induced subgraphs with respect to the number of components

Abstract. Inspired by the study of community structure in connection networks, we introduce the graph polynomial Q (G; x, y), the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of Q (G; x, y) using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate Q (G; x, y) to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimination polynomial introduced by I. Averbouch, B. Godlin and J.A. Makowsky (2008). We show that Q(G; x, y) is vertex reconstructible in the sense of Kelly and Ulam, discuss its use in computing residual connectedness reliability. Finally we show that the computation of Q(G; x, y) is ♯P-hard, but Fixed Paramete