Related papers: Defensive alliance polynomial

A graph is said to be well-dominated if all its minimal dominating sets are of the same size. The class of well-dominated graphs forms a subclass of the well studied class of well-covered graphs. While the recognition problem for the class…

Chevalley's theorem states that for any simple finite dimensional Lie algebra G (1) the restriction homomorphism of the algebra of polynomials on G onto the Cartan subalgebra H induces an isomorphism between the algebra of G-invariant…

A graph $G$ with vertex set $V(G)$ and edge set $E(G)$ is said to be word-representable if there exists a word $w$ over the alphabet $V(G)$ such that, for any two distinct letters $x,y \in V(G)$, the letters $x$ and $y$ alternate in $w$ if…

Let X $\subseteq$ {0, 1} n. Then the daisy cube Q n (X) is introduced as the sub-graph of Q n induced by the intersection of the intervals I(x, 0 n) over all x $\in$ X. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas…

We propose bipartite analogues of comparability and cocomparability graphs. Surprizingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex…

Let G = (V;E) be a simple graph. We consider domination polynomial, matching polynomial and edge cover polynomial of G. Graphs which their polynomials have few roots can give sometimes a very surprising information about the structure of…

A set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbors is in S. The problem of finding a defensive alliance of minimum size in a given graph is NP-hard and there are polynomial-time…

We show that the dominance complex $\mathcal{D}(G)$ of a graph $G$ coincides with the combinatorial Alexander dual of the neighborhood complex $\mathcal{N}(\overline{G})$ of the complement of $G$. Using this, we obtain a relation between…

In this paper we investigate the $directed$ $normalizing$ $graph$ associated with a group $G$, defined as the simple directed graph whose vertices are the elements of $G$, with an arrow from $x$ to $y$ whenever the subgroup $\langle x…

We study the dual graph polynomials and the case when a Feynman graph has no triangles but has a 4-face. This leads to the proof of the duality-admissibility of all graphs up to 18 loops. As a consequence, the $c_2$ invariant is the same…

Deep learning on graph structures has shown exciting results in various applications. However, few attentions have been paid to the robustness of such models, in contrast to numerous research work for image or text adversarial attack and…

Tutte's dichromate T(x,y) is a well known graph invariant. Using the original definition in terms of internal and external activities as our point of departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In the…

A connected matching in a graph G consists of a set of pairwise disjoint edges whose covered vertices induce a connected subgraph of G. While finding a connected matching of maximum cardinality is a well-solved problem, it is NP-hard to…

In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having…

A set $S\subseteq V$ in an isolate-free graph $G$ is a total restrained dominating set, abbreviated TRD-set, if every vertex in $V$ is adjacent to a vertex in $S$, and every vertex in $V\setminus S$ is adjacent to a vertex in $V\setminus…

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $\gamma(G)$ is the domination…

\noindent A paired coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a paired dominating set but whose union $V_1 \cup V_2$ is a paired dominating set. A paired coalition partition…

In this paper we characterize "large" regular graphs using certain entries in the projection matrices onto the eigenspaces of the graph. As a corollary of this result, we show that "large" association schemes become $P$-polynomial…

Polynomials on stranded graphs are higher dimensional generalization of Tutte and Bollob\'as-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we deepen the analysis of the polynomial invariant defined on rank 3 weakly-colored…

Some of the most common mathematical models in biology, chemistry, physics, and engineering, are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand sides. Inspired by notions and results that…