1. CMB Online first
 Khojasteh, Sohiela; Nikmehr, Mohammad Javad

The Weakly Nilpotent Graph of a Commutative Ring
Let $R$ be a commutative ring with nonzero identity. In this
paper, we introduced the weakly nilpotent graph of a commutative
ring. The weakly nilpotent graph of $R$ is denoted by $\Gamma_w(R)$
is a graph with the vertex set $R^{*}$ and two vertices $x$ and
$y$ are adjacent if and only if $xy\in N(R)^{*}$, where $R^{*}=R\setminus\{0\}$
and $N(R)^{*}$ is the set of all nonzero nilpotent elements
of $R$. In this article, we determine the diameter of weakly
nilpotent graph of an Artinian ring. We prove that if $\Gamma_w(R)$
is a forest, then $\Gamma_w(R)$ is a union of a star and some
isolated vertices. We study the clique number, the chromatic
number and the independence number of $\Gamma_w(R)$. Among other
results, we show that for an Artinian ring $R$, $\Gamma_w(R)$
is not a disjoint union of cycles or a unicyclic graph. For Artinan
ring, we determine $\operatorname{diam}(\overline{\Gamma_w(R)})$. Finally, we
characterize all commutative rings $R$ for which $\overline{\Gamma_w(R)}$
is a cycle, where $\overline{\Gamma_w(R)}$ is the complement
of the weakly nilpotent graph of $R$.
Keywords:weakly nilpotent graph, zerodivisor graph, diameter, girth Categories:05C15, 16N40, 16P20 

2. CMB 2016 (vol 60 pp. 26)
3. CMB 2016 (vol 60 pp. 3)
 Akbari, Saeeid; Alilou, Abbas; Amjadi, Jafar; Sheikholeslami, Seyed Mahmoud

The Coannihilatingideal Graphs of Commutative Rings
Let $R$ be a commutative ring with identity. The
coannihilatingideal graph of $R$, denoted by $\mathcal{A}_R$,
is
a graph whose vertex set is the set of all nonzero proper ideals
of $R$ and two distinct vertices $I$ and $J$ are adjacent
whenever ${\operatorname {Ann}}(I)\cap {\operatorname {Ann}}(J)=\{0\}$. In this paper we
initiate the study of the coannihilating ideal graph of a
commutative ring and we investigate its properties.
Keywords:commutative ring, coannihilating ideal graph Categories:13A15, 16N40 

4. CMB 2016 (vol 60 pp. 12)
 Akbari, Saieed; Miraftab, Babak; Nikandish, Reza

Comaximal Graphs of Subgroups of Groups
Let $H$ be a group. The comaximal graph of subgroups
of $H$, denoted by $\Gamma(H)$, is a
graph whose vertices are nontrivial and proper subgroups of
$H$ and two distinct vertices $L$
and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In
this paper, we study the connectivity, diameter, clique number
and vertex
chromatic number of $\Gamma(H)$. For instance, we show that
if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$
is connected with diameter at most $3$. Also, we characterize
all finite groups whose comaximal graphs are connected.
Among other results, we show that if $H$ is a finitely generated
solvable group and $\Gamma(H)$ is connected and moreover the
degree of a maximal subgroup is finite, then $H$ is finite.
Furthermore, we show that the degree of each vertex in the
comaximal graph of a general linear group over an algebraically
closed field is zero or infinite.
Keywords:comaximal graphs of subgroups of groups, diameter, nilpotent group, solvable group Categories:05C25, 05E15, 20D10, 20D15 

5. CMB 2016 (vol 60 pp. 43)
6. CMB 2016 (vol 60 pp. 197)
 Tang, Zikai; Deng, Hanyuan

Degree Kirchhoff Index of Bicyclic Graphs
Let $G$ be a connected graph with vertex set $V(G)$. The degree
Kirchhoff index of $G$ is defined as $S'(G) =\sum_{\{u,v\}\subseteq
V(G)}d(u)d(v)R(u,v)$, where $d(u)$ is the degree of vertex $u$,
and
$R(u, v)$ denotes the resistance distance between vertices $u$
and
$v$. In this paper, we characterize the graphs having maximum
and
minimum degree Kirchhoff index among all $n$vertex bicyclic
graphs
with exactly two cycles.
Keywords:degree Kirchhoff index, resistance distance, bicyclic graph, extremal graph Categories:05C12, 05C35 

7. CMB 2016 (vol 59 pp. 806)
 Izumiya, Shyuichi

Geometric Interpretation of Lagrangian Equivalence
As an application of the theory of
graphlike Legendrian unfoldings, relations of the hidden structures
of caustics and wave front propagations are revealed.
Keywords:wave front propagations, big wave fronts, graphlike Legendrian unfoldings, caustics Categories:58K05, 57R45, 58K60 

8. CMB 2016 (vol 60 pp. 206)
 Vetrik, Tomáš

The Metric Dimension of Circulant Graphs
A subset $W$ of the vertex set of a graph $G$ is called a resolving
set of $G$ if for every pair of distinct vertices $u, v$ of $G$,
there is $w \in W$ such that the~distance of $w$ and $u$ is different
from the distance of $w$ and $v$. The~cardinality of a~smallest
resolving set is called the metric dimension of $G$, denoted
by $dim(G)$. The circulant graph $C_n (1, 2, \dots , t)$ consists
of the vertices $v_0, v_1, \dots , v_{n1}$ and the~edges $v_i
v_{i+j}$, where $0 \le i \le n1$, $1 \le j \le t$ $(2 \le t
\le \lfloor \frac{n}{2} \rfloor)$, the indices are taken modulo
$n$. Grigorious et al. [On the metric dimension of circulant
and Harary graphs, Applied Mathematics and Computation 248 (2014),
4754] proved that $dim(C_n (1,2, \dots , t))
\ge t+1$ for $t \lt \lfloor \frac{n}{2} \rfloor$, $n \ge 3$, and they
presented a~conjecture saying that $dim(C_n (1,2, \dots , t))
= t+p1$ for $n=2tk+t+p$, where $3 \le p \le t+1$. We disprove
both statements. We show that if $t \ge 4$ is even, there exists
an infinite set of values of $n$ such that $dim(C_n (1,2, \dots
, t)) = t$. We also prove that $dim(C_n (1,2, \dots , t)) \le
t + \frac{p}{2}$ for $n=2tk+t+p$, where $t$ and $p$ are even,
$t \ge 4$, $2 \le p \le t$ and $k \ge 1$.
Keywords:metric dimension, resolving set, circulant graph, Cayley graph, distance Categories:05C35, 05C12 

9. CMB 2016 (vol 59 pp. 794)
 Hashemi, Ebrahim; Amirjan, R.

Zerodivisor Graphs of Ore Extensions over Reversible Rings
Let $R$ be an associative ring with identity.
First we prove some results about zerodivisor graphs of reversible
rings. Then we study the zerodivisors of the skew power series
ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$compatible. Moreover, we compare the diameter and girth of the zerodivisor
graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$,
when
$R$ is reversible and $(\alpha,\delta)$compatible.
Keywords:zerodivisor graphs, reversible rings, McCoy rings, polynomial rings, power series rings Categories:13B25, 05C12, 16S36 

10. CMB 2016 (vol 59 pp. 652)
 Su, Huadong

On the Diameter of Unitary Cayley Graphs of Rings
The unitary Cayley graph of a ring $R$, denoted
$\Gamma(R)$, is the simple graph
defined on all elements of $R$, and where two vertices $x$ and
$y$
are adjacent if and only if $xy$ is a unit in $R$. The largest
distance between all pairs of vertices of a graph $G$ is called
the
diameter of $G$, and is denoted by ${\rm diam}(G)$. It is proved
that for each integer $n\geq1$, there exists a ring $R$ such
that
${\rm diam}(\Gamma(R))=n$. We also show that ${\rm
diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$
selfinjective and classify all those rings with ${\rm
diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively.
Keywords:unitary Cayley graph, diameter, $k$good, unit sum number, selfinjective ring Categories:05C25, 16U60, 05C12 

11. CMB 2016 (vol 59 pp. 705)
 Chen, Yichao; Yin, Xuluo

The Thickness of the Cartesian Product of Two Graphs
The thickness of a graph $G$ is the minimum number
of planar subgraphs whose union is $G.$ A
$t$minimal graph is a graph of thickness $t$ which contains
no proper subgraph of thickness $t.$ In this paper, upper and
lower bounds are obtained for the thickness, $t(G\Box H)$, of
the Cartesian
product of two graphs $G$ and $H$, in terms of the thickness
$t(G)$ and $t(H)$.
Furthermore, the thickness of the Cartesian product of two planar
graphs and of a $t$minimal graph and a planar graph are determined.
By using a new planar decomposition of the complete bipartite
graph $K_{4k,4k},$ the thickness of the Cartesian product of
two complete bipartite graphs $K_{n,n}$ and $K_{n,n}$ is also
given, for $n\neq 4k+1$.
Keywords:planar graph, thickness, Cartesian product, $t$minimal graph, complete bipartite graph Category:05C10 

12. CMB 2016 (vol 59 pp. 641)
 Shaveisi, Farzad

Some Results on the Annihilatingideal Graphs
The annihilatingideal graph
of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a
graph whose vertex set consists of all nonzero annihilating
ideals and two distinct
vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here,
we show that if $R$ is a reduced ring and the independence
number of $\mathbb{AG}(R)$ is finite, then the edge chromatic
number of $\mathbb{AG}(R)$ equals its maximum degree
and this number equals $2^{{\rm Min}(R)1}1$; also, it is
proved that the independence number of $\mathbb{AG}(R)$ equals
$2^{{\rm Min}(R)1}$, where ${\rm Min}(R)$ denotes the set
of minimal prime ideals of $R$.
Then we give some criteria for a graph to be isomorphic with
an annihilatingideal graph of a ring.
For example, it is shown that every bipartite annihilatingideal
graph is a complete bipartite graph with at most two horns. Among
other results, it is shown that a finite graph $\mathbb{AG}(R)$
is not Eulerian, and it is Hamiltonian if and only if $R$ contains
no Gorenstain ring as its direct summand.
Keywords:annihilatingideal graph, independence number, edge chromatic number, bipartite, cycle Categories:05C15, 05C69, 13E05, 13E10 

13. CMB 2016 (vol 59 pp. 748)
 Dolžan, David

The Metric Dimension of the Total Graph of a Finite Commutative Ring
We study the total graph of a finite commutative ring. We calculate
its metric dimension in the case when the Jacobson radical of
the ring is nontrivial and we examine the metric dimension of
the total graph of a product of at most two fields, obtaining
either exact values in some cases or bounds in other, depending
on the number of elements in the respective fields.
Keywords:total graph, finite ring, metric dimension Categories:13M99, 05E40 

14. CMB 2016 (vol 59 pp. 440)
 Zhang, Haihui

A Note on 3choosability of Planar Graphs Related to Montanssier's Conjecture
A graph $G=(V,E)$ is $L$colorable if for a given list
assignment $L=\{L(v):v\in V(G)\}$, there exists a proper coloring
$c$ of $G$ such that $c(v)\in L(v)$ for all $v\in V$. If $G$ is
$L$colorable for every list assignment $L$ with $L(v)\geq
k$ for
all $v\in V$, then $G$ is said to be $k$choosable. Montassier
(Inform. Process. Lett. 99 (2006) 6871) conjectured that every
planar
graph without cycles of length 4, 5, 6, is 3choosable. In this
paper,
we prove that every planar graph without 5, 6 and 10cycles,
and
without two triangles at distance less than 3 is 3choosable.
Keywords:choosability, planar graph, cycle Category:05C15 

15. CMB 2016 (vol 59 pp. 287)
 Dukes, Peter; Lamken, E.R.; Ling, Alan C.H.

An Existence Theory for Incomplete Designs
An incomplete pairwise balanced design is equivalent to a pairwise
balanced design with a distinguished block, viewed as a `hole'.
If there are $v$ points, a hole of size $w$, and all (other)
block sizes equal $k$, this is denoted IPBD$((v;w),k)$. In addition
to congruence restrictions on $v$ and $w$, there is also a necessary
inequality: $v \gt (k1)w$. This article establishes two main existence
results for IPBD$((v;w),k)$: one in which $w$ is fixed and $v$
is large, and the other in the case $v \gt (k1+\epsilon) w$ when
$w$ is large (depending on $\epsilon$). Several possible generalizations
of the problem are also discussed.
Keywords:block design, hypergraph Category:05C70 

16. CMB 2015 (vol 59 pp. 50)
17. CMB 2015 (vol 59 pp. 3)
 Alfuraidan, Monther Rashed

The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph
We study the existence of fixed points for contraction multivalued
mappings in modular metric spaces endowed with a graph. The
notion of a modular metric on an arbitrary set and the corresponding
modular spaces, generalizing classical modulars over linear spaces
like Orlicz spaces, were recently introduced. This paper can
be seen as a generalization of Nadler's and Edelstein's fixed
point theorems to modular metric spaces endowed with a graph.
Keywords:fixed point theory, modular metric spaces, multivalued contraction mapping, connected digraph. Categories:47H09, 46B20, 47H10, 47E10 

18. CMB 2015 (vol 59 pp. 95)
 Gonçalves, Daniel; Li, Hui; Royer, Danilo

Faithful Representations of Graph Algebras via Branching Systems
We continue to investigate branching systems of directed graphs
and their connections with graph algebras. We give a sufficient
condition under which the representation induced from a branching
system of a directed graph is faithful and construct a large
class of branching systems that satisfy this condition. We finish
the paper by providing a proof of the converse of the CuntzKrieger
uniqueness theorem for graph algebras by means of branching systems.
Keywords:C*algebra, graph algebra, Leavitt path algebra, branching system, representation Categories:46L05, 37A55 

19. CMB 2015 (vol 58 pp. 610)
20. CMB 2015 (vol 58 pp. 320)
 Llamas, Aurora; MartínezBernal, José

Cover Product and Betti Polynomial of Graphs
For disjoint graphs $G$ and $H$, with fixed
vertex covers
$C(G)$ and $C(H)$, their cover product is the graph $G
\circledast
H$ with vertex set
$V(G)\cup V(H)$ and edge set $E(G)\cup E(H)\cup\{\{i,j\}:i\in
C(G), j\in
C(H)\}$. We describe the graded Betti numbers of $G\circledast
H$ in terms of those of
$G$ and $H$. As applications we obtain: (i) For any positive
integer $k$ there
exists a connected bipartite graph $G$ such that $\operatorname{reg}
R/I(G)=\mu_S(G)+k$, where,
$I(G)$ denotes the edge ideal of $G$, $\operatorname{reg} R/I(G)$
is the CastelnuovoMumford
regularity of $R/I(G)$ and $\mu_S(G)$ is the induced or strong
matching number of
$G$; (ii) The graded Betti numbers of the complement of a tree
only depends upon
its number of vertices; (iii) The $h$vector of $R/I(G\circledast
H)$ is described in
terms of the $h$vectors of $R/I(G)$ and $R/I(H)$. Furthermore,
in a different
direction, we give a recursive formula for the graded Betti numbers
of chordal
bipartite graphs.
Keywords:CastelnuovoMumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial ideal Categories:13D02, 05E45 

21. CMB 2015 (vol 58 pp. 271)
22. CMB 2015 (vol 58 pp. 306)
 Khoshkhah, Kaveh; Zaker, Manouchehr

On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold
Let $G$ be a graph and $\tau$ be an assignment of nonnegative
integer thresholds to the vertices of $G$. A subset of vertices,
$D$ is said to be a $\tau$dynamic monopoly, if $V(G)$ can be
partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$
and for any $i\in \{0, \ldots, k1\}$, each vertex $v$ in $D_{i+1}$
has at least $\tau(v)$ neighbors in $D_0\cup \ldots \cup D_i$.
Denote the size of smallest $\tau$dynamic monopoly by $dyn_{\tau}(G)$
and the average of thresholds in $\tau$ by $\overline{\tau}$.
We show that the values of $dyn_{\tau}(G)$ over all assignments
$\tau$ with the same average threshold is a continuous set of
integers. For any positive number $t$, denote the maximum $dyn_{\tau}(G)$
taken over all threshold assignments $\tau$ with $\overline{\tau}\leq
t$, by $Ldyn_t(G)$. In fact, $Ldyn_t(G)$ shows the worstcase
value of a dynamic monopoly when the average threshold is a given
number $t$. We investigate under what conditions on $t$, there
exists an upper bound for $Ldyn_{t}(G)$ of the form $cG$, where
$c\lt 1$. Next, we show that $Ldyn_t(G)$ is coNPhard for planar
graphs but has polynomialtime solution for forests.
Keywords:spread of influence in graphs, irreversible dynamic monopolies, target set selection Categories:05C69, 05C85 

23. CMB 2015 (vol 58 pp. 317)
24. CMB 2014 (vol 58 pp. 105)
 HosseinZadeh, Samaneh; Iranmanesh, Ali; Hosseinzadeh, Mohammad Ali; Lewis, Mark L.

On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups
The prime vertex graph, $\Delta (X)$, and the common divisor graph,
$\Gamma (X)$, are two graphs that have been defined on a set of
positive integers $X$.
Some
properties of these graphs have been studied in the cases where either
$X$ is the set of character degrees of a group or $X$ is the set of
conjugacy class sizes of a group. In this paper, we gather some
results on these graphs arising in the context of direct product of
two groups.
Keywords:prime vertex graph, common divisor graph, character degree, class sizes, graph operation Categories:20E45, 05C25, 05C76 

25. CMB 2014 (vol 58 pp. 150)
 Ostrovskii, Mikhail I.

Connections Between Metric Characterizations of Superreflexivity and the RadonNikodÃ½ Property for Dual Banach Spaces
Johnson and Schechtman (2009)
characterized superreflexivity in terms of finite diamond graphs.
The present author characterized the RadonNikodÃ½m property
(RNP) for dual spaces in terms of the infinite diamond. This
paper
is devoted to further study of relations between metric
characterizations of superreflexivity and the RNP for dual spaces.
The main result is that finite subsets of any set $M$ whose
embeddability characterizes the RNP for dual spaces, characterize
superreflexivity. It is also observed that the converse statement
does not hold, and that $M=\ell_2$ is a counterexample.
Keywords:Banach space, diamond graph, finite representability, metric characterization, RadonNikodÃ½m property, superreflexivity Categories:46B85, 46B07, 46B22 
