arXiv subjects
Publications and source records attributed to S. Akbari.
We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on 2n vertices. The upper bound is sharp for n even. For n odd we state a conjecture on a sharp upper bound.
Let $G$ be a graph. A good function is a function $f:V(G)\rightarrow \{-1,1\}$, satisfying $f(N(v))\geq 1$, for each $v\in V(G)$, where $ N(v)=\{u\in V(G)\, |\, uv\in E(G) \} $ and $f(S) = \sum_{u\in S} f(u)$ for every $S \subseteq V(G) $. For every cubic graph $G$ of order $ n, $ we prove that $ γ(G) \leq \frac{5n}{7} $ and show that this inequality is sharp. A function $f:V(G)\rightarrow \{-1,1\}$ is called a nice function, if $f(N[v])\le1$, for each $v\in V(G)$, where $ N[v]=\{v\} \cup N(v) $. Define $\overlineβ(G)=max\{f(V(G))\}$, where $f$ is a nice function for $G$. We show that $\overlineβ(G)\ge -\frac{3n}{7}$ for every cubic graph $G$ of order $n$, which improves the best known bound $-\frac{n}{2}$.
Let $R$ be a ring (not necessary commutative) with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is a graph with elements of $R$ as its vertices and two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a unit element of $R$. It was proved that if $R$ is a commutative ring and $\fm$ is a maximal ideal of $R$ such that $|R/\fm|=2$, then $G(R)$ is a complete bipartite graph if and only if $(R, \fm)$ is a local ring. In this paper we generalize this result by showing that if $R$ is a ring (not necessary commutative), then $G(R)$ is a complete $r$-partite graph if and only if $(R, \fm)$ is a local ring and $r=|R/m|=2^n$, for some $n \in \N$ or $R$ is a finite field. Among other results we show that if $R$ is a left Artinian ring, $2 \in U(R)$ and the clique number of $G(R)$ is finite, then $R$ is a finite ring.
Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a graph $Γ_G$ (called non-commuting graph of $G$) with $G$ as follows: take $G\setminus Z(G)$ as the vertices of $Γ_G$ and join two distinct vertices $x$ and $y$, whenever $xy\neq yx$. Here, we prove that "the commutativity pattern of a finite non-abelian $p$-group determine its order among the class of groups"; this means that if $P$ is a finite non-abelian $p$-group such that $Γ_P\cong Γ_H$ for some group $H$, then $|P|=|H|$.
Let $G$ be a graph. A zero-sum flow in $G$ is an assignment of nonzero real number to the edges such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be naturel number. A zero-sum $k$-flow is a flow with value from the set $\{\pm 1, \pm 2,..., \pm(k-1)\}$. It has been conjectured that every $r$-regular graph, $r\geq 3$, admits a zero-sum 5-flow. In this paper we give an affirmative answer to this conjecture, exept for r=5.
Let $R$ be a ring with unity. The graph $Γ(R)$ is a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let $Γ_2(R)$ is the subgraph of $Γ(R)$ induced by the non-unit elements. H.R. Maimani et al. [H.R. Maimani et al., Comaximal graph of commutative rings, J. Algebra $319$ $(2008)$ $1801$-$1808$] proved that: ``If $R$ is a commutative ring with unity and the graph $Γ_2(R)\backslash J(R)$ is $n$-partite, then the number of maximal ideals of $R$ is at most $n$." The proof of this result is not correct. In this paper we present a correct proof for this result. Also we generalize some results given in the aforementioned paper for the non-commutative rings.
We conjecture that the sum of the square roots of the Laplacian eigenvalues of a graph does not exceed the sum of the square roots of its signless Laplacian eigenvalues.
Let G be a graph. It was proved that if G is a planar graph without {4, 6, 7}-cycles and without two 5-cycles sharing exactly one edge, then G 3-colorable. We observed that the proof of this result is not correct.
In this paper we introduce a new class of codes for over-loaded synchronous wireless and optical CDMA systems which increases the number of users for fixed number of chips without introducing any errors. Equivalently, the chip rate can be reduced for a given number of users, which implies bandwidth reduction for downlink wireless systems. An upper bound for the maximum number of users for a given number of chips is derived. Also, lower and upper bounds for the sum channel capacity of a binary over-loaded CDMA are derived that can predict the existence of such over-loaded codes. We also propose a simplified maximum likelihood method for decoding these types of over-loaded codes. Although a high percentage of the over-loading factor degrades the system performance in noisy channels, simulation results show that this degradation is not significant. More importantly, for moderate values of Eb/N0 (in the range of 6-10 dB) or higher, the proposed codes perform much better than the binary Welch bound equality sequences.
The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $G$ be a graph of order $n$ and ${\rm rank}(G)$ be the rank of the adjacency matrix of $G$. In this paper we characterize all graphs with $E(G)={\rm rank}(G)$. Among other results we show that apart from a few families of graphs, $E(G)\geq 2\max(χ(G), n-χ(\bar{G}))$, where $n$ is the number of vertices of $G$, $\bar{G}$ and $χ(G)$ are the complement and the chromatic number of $G$, respectively. Moreover some new lower bounds for $E(G)$ in terms of ${\rm rank}(G)$ are given.