Brooks proved that the chromatic number of a loopless connected graph G is at most the maximum degree of G unless G is an odd cycle or a clique. This note proves an analogue of this theorem for GF (p)-representable matroids when p is prime, thereby verifying a natural generalization of a conjecture of Peter Nelson.

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph Kn on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G H of graphs. As the main result of this paper, we prove that η(G1 G2) ≥ h √ l (1− o(1)) for any two g...

Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of #pHomsToH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating t...

Let $G$ be a finite group. The prime graph of $G$ is a graph $Gamma(G)$ with vertex set $pi(G)$, the set of all prime divisors of $|G|$, and two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. In this paper we prove that if $Gamma(G)=Gamma(G_2(5))$, then $G$ has a normal subgroup $N$ such that $pi(N)subseteq{2,3,5}$ and $G/Nequiv G_2(5)$.

a multicone graph is defined to be join of a clique and a regular graph. a graph $ g $ is  cospectral with  graph $ h $ if their adjacency matrices have the same eigenvalues. a graph $ g $ is  said to be  determined by its spectrum or ds for short, if for any graph $ h $ with $ spec(g)=spec(h)$, we conclude that $ g $ is isomorphic to $ h $. in this paper, we present new classes of  multicone g...

We s(ate the following conjecture and prove it for the case where q is a proper prime power: Let A be a nonsingular n by n matrix over the finite fieM GF~, q~-4, then there exists a vector x in (GFa) ~ such that both x and ~4x have no zero component. In this note we consider-the following conjecture: Conjecture 1. Let A be a nonsingular n by n matrix over the finite field GFa, q~_4, then there ...

in this paper we will prove that the simple group g2(q) where 2 < q = 1(mod3)is recognizable by the set of its order components, also other word we prove that if g is anite group with oc(g) = oc(g2(q)), then g is isomorphic to g2(q).

Let G be a finite abelian group of order n. For any subset B of G with B = −B, the Cayley graph GB is a graph on vertex set G in which ij is an edge if and only if i − j ∈ B. It was shown by Ben Green [3] that when G is a vector space over a finite field Z/pZ, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than c logn log logn, where c > ...

The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

The Harmonic index $ H(G) $ of a graph $ G $ is defined as the sum of the weights $ dfrac{2}{d(u)+d(v)} $ of all edges $ uv $ of $G$, where $d(u)$ denotes the degree of the vertex $u$ in $G$. In this work, we prove the conjecture $dfrac{H(G)}{D(G)} geq dfrac{1}{2}+dfrac{1}{3(n-1)}&nbsp; $ given by Jianxi Liu in 2013 when G is a unicyclic graph and give a better bound $ dfrac{H(G)}{D(G)}geq dfra...