### فهرست مطالب

• Volume:1 Issue:2, 2016
• تاریخ انتشار: 1395/06/20
• تعداد عناوین: 6
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• Behrouz Kheirfam * Pages 83-102
In this paper, we present a full Newton step feasible interior-point method for circular cone optimization by using Euclidean Jordan algebra. The search direction is based on the Nesterov-Todd scaling scheme, and only full-Newton step is used at each iteration. Furthermore, we derive the iteration bound that coincides with the currently best known iteration bound for small-update methods.
Keywords: Circular cone optimization, Full-Newton step, Interior-point methods, Euclidean Jordan algebra
• Vladimir Samodivkin * Pages 103-116
For a graph $G$ let $gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets, but $G-v$ has a unique minimum dominating set for each $vin V(G)$. We show that each hypo-$mathcal{UD}$ graph $G$ of order at least $3$ is connected and $gamma(G-v) < gamma(G)$ for all $v in V$. We obtain a tight upper bound on the order of a hypo-$mathcal{P}$ graph in terms of the domination number and maximum degree of the graph, where $mathcal{P} in {mathcal{UD}, mathcal{ED}}$. Families of circulant graphs, which achieve these bounds, are presented. We also prove that the bondage number of any hypo-$mathcal{UD}$ graph is not more than the minimum degree plus one.
Keywords: domination number, efficient domination, unique domination, hypo-property
• Abbas Alilou, Jafar Amjadi * Pages 117-135
Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$ is called an annihilating ideal if there exists $r\in R\setminus \{0\}$ such that $Ir=(0)$ and an ideal $I$ of $R$ is called an essential ideal if $I$ has non-zero intersection with every other non-zero ideal of $R$. The sum-annihilating essential ideal graph of $R$, denoted by $\mathcal{AE}_R$, is a graph whose vertex set is the set of all non-zero annihilating ideals and two vertices $I$ and $J$ are adjacent whenever ${\rm Ann}(I)+{\rm Ann}(J)$ is an essential ideal. In this paper we initiate the study of the sum-annihilating essential ideal graph. We first characterize all rings whose sum-annihilating essential ideal graph are stars or complete graphs and then establish sharp bounds on domination number of this graph. Furthermore determine all isomorphism classes of Artinian rings whose sum-annihilating essential ideal graph has genus zero or one.
Keywords: Commutative rings, annihilating ideal, essential ideal, genus of a graph
• Mehdi Eliasi *, Ali Ghalavand Pages 137-148
For a graph $G$ with edge set $E(G)$, the multiplicative sum Zagreb index of $G$ is defined as $Pi^*(G)=Pi_{uvin E(G)}[d_G(u)+d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$. In this paper, we first introduce some graph transformations that decrease this index. In application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum Zagreb indeces among all trees of order $ngeq 13$.
Keywords: Multiplicative Sum Zagreb Index, Graph Transformation, Branching Point, trees
• Maryam Atapour *, Abdollah Khodkar Pages 149-164
Let $D=(V,A)$ be a finite simple directed graph. A function $f:Vlongrightarrow {-1,0,1}$ is called a twin minus dominating function (TMDF) if $f(N^-[v])ge 1$ and $f(N^+[v])ge 1$ for each vertex $vin V$. The twin minus domination number of $D$ is $gamma_{-}^*(D)=min{w(f)mid f mbox{ is a TMDF of } D}$. In this paper, we initiate the study of twin minus domination numbers in digraphs and present some lower bounds for $gamma_{-}^*(D)$ in terms of the order, size and maximum and minimum in-degrees and out-degrees.
Keywords: twin domination in digraphs, minus domination in graphs, twin minus domination in digraphs
• Nasrin Dehgardi *, Lutz Volkmann Pages 165-178
Let $D$ be a finite and simple digraph with vertex set $V(D)$‎. ‎A signed total Roman $k$-dominating function (STR$k$DF) on‎ ‎$D$ is a function $f:V(D)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎ ‎that (i) $sum_{xin N^{-}(v)}f(x)ge k$ for each‎ ‎$vin V(D)$‎, ‎where $N^{-}(v)$ consists of all vertices of $D$ from‎ ‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎ ‎$f(u)=-1$ has an inner neighbor $v$ for which $f(v)=2$‎. ‎The weight of an STR$k$DF $f$ is $omega(f)=sum_{vin V (D)}f(v)$‎. ‎The signed total Roman $k$-domination number $gamma^{k}_{stR}(D)$‎ ‎of $D$ is the minimum weight of an STR$k$DF on $D$‎. ‎In this paper we‎ ‎initiate the study of the signed total Roman $k$-domination number‎ ‎of digraphs‎, ‎and we present different bounds on $gamma^{k}_{stR}(D)$‎. ‎In addition‎, ‎we determine the signed total Roman $k$-domination‎ ‎number of some classes of digraphs‎. ‎Some of our results are extensions‎ ‎of known properties of the signed total Roman $k$-domination‎ ‎number $gamma^{k}_{stR}(G)$ of graphs $G$‎.
Keywords: Digraph‎, ‎Signed total Roman k-dominating function‎, ‎Signed total ‎Rom‎an k-domination‎