Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree o...

We describe a sufficient condition for graphs used in a construction of Thomassen (which yields hypohamiltonian graphs) to produce maximally non-hamiltonian (MNH) graphs as well. Then we show that the Coxeter graph fulfils this sufficient condition, and thus applying the Thomassen’s construction to multiple copies of the Coxeter graph yields infinitely many MNH graphs with girth 7. So far, the ...

We conjecture that every planar graph of odd-girth at least 11 admits a homomorphism to the Coxeter graph. Supporting this conjecture, we prove that every planar graph of odd-girth at least 17 admits a homomorphism to the Coxeter graph.

In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least 7. We show that the domination number of a connected graph with girth at least 7 is either 1 or at least 1 2 (3 + √ 8(m− n) + 9), where n is the number of vertices in the graph and m is the number of edges in the graph. For graphs with minimum degree 2 and girth at least 7, the lower bound c...

A star-factor of a graph G is a spanning subgraph of G such that each component of which is a star. Recently, Hartnell and Rall studied a family U of graphs satisfying the property that every star-factor of a member graph has the same number of edges. They determined the family U when the girth is at least five. In this paper, we investigate the family of graphs with girth three and determine a...

We prove that a dense subgroup of Homeo+(I) is not elementary amenable and (if it is finitely generated) has infinite girth. We also show that the topological group Homeo+(I) does not satisfy the Stability of the Generators Property, moreover, any finitely subgroup of Homeo+(I) admits a faithful discrete representation in it.

for any $k in mathbb{n}$, the $k$-subdivision of graph $g$ is a simple graph $g^{frac{1}{k}}$, which is constructed by replacing each edge of $g$ with a path of length $k$. in [moharram n. iradmusa, on colorings of graph fractional powers, discrete math., (310) 2010, no. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $g$ has been introduced as a fractional power of $g$, denoted by ...

We prove the Girth Alternative for finitely generated subgroups of PLo(I). We also prove that a finitely generated subgroup of Homeo+(I) which is sufficiently rich with hyperbolic-like elements has infinite girth.

abstract. let $l$ be a lattice with the least element $0$. an element $xin l$ is a zero divisor if $xwedge y=0$ for some $yin l^*=lsetminus left{0right}$. the set of all zero divisors is denoted by $z(l)$. we associate a simple graph $gamma(l)$ to $l$ with vertex set $z(l)^*=z(l)setminus left{0right}$, the set of non-zero zero divisors of $l$ and distinct $x,yin z(l)^*$ are adjacent if and only...