As the extension of the previous work by Ciucu and the present authors (J. Combin. Theory Ser. A 112(2005) 105–116), this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of ...

In this article we compute analytically the number of Nash Equilibria (NE) for a two-choice game played on (circular) ladder graph with $2n$ players. We consider set games generic payoff parameters, only requirement that NE occurs if players choose opposite strategies (anti-coordination game). The results show both, and circular ladder, grows exponentially (half) $n$, as $N_{NE}(2n)\sim C(\varp...

the concept of the bipartite divisor graph for integer subsets has been considered in [‎graph combinator., ‎ 26 (2010) 95--105‎.]. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g$ is a solvable group‎, ‎then the number of connected components of this graph is at most $2$ and if $g...

in this paper we construct antimagic labelings of the regular complete multipartite graphs and we also extend the construction to some families of regular graphs.

A ranking on a graph is an assignment of positive integers to its vertices such that any path between two vertices with the same label contains a vertex with a larger label. The rank number of a graph is the fewest number of labels that can be used in a ranking. The rank number of a graph is known for many families, including the ladder graph P2 × Pn. We consider how ”bending” a ladder affects ...

A graph G∗ is 1-edge fault-tolerant with respect to a graph G, denoted by 1-EFT(G), if every graph obtained by removing any edge from G∗ contains G. A 1-EFT(G) graph is optimal if it contains the minimum number of edges among all 1-EFT(G) graphs. The kth ladder graph, Lk , is defined to be the cartesian product of the Pk and P2 where Pn is the n-vertex path graph. In this paper, we present seve...

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This “density conjecture” has been proved for the underlying graph Z. We ...

A popular theory of self-organized criticality predicts that the stationary density of the Abelian sandpile model equals the threshold density of the corresponding fixed-energy sandpile. We recently announced that this "density conjecture" is false when the underlying graph is any of Z2, the complete graph K(n), the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. In this...

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...