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## Publications

Publications (110)

Let G be a graph. The energy E(G) is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. In [Energy, matching number and odd cycles of graphs, Linear Algebra Appl. 577 (2019) 159-167] it has been proved that for a graph G whose cycles are odd and vertex disjoint, if from each cycle of G, we remove an arbitrary edge to ob...

Let G be a graph and I be an interval. In this paper, we present bounds for the number mGI of Laplacian eigenvalues in I in terms of structural parameters of G. In particular, we show that mG(n−α(G),n]≤n−α(G) and mG(n−d(G)+3,n]≤n−d(G)−1, where α(G) and d(G) denote the independence number and the diameter of G, respectively. Also, we characterize bi...

In 2011, Haemers asked the following question: If S is the Seidel matrix of a graph of order n and S is singular, does there exist an eigenvector of S corresponding to 0 which has only ±1 elements?
In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every n...

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N(D)=∑i=1nσi, where σ1≥⋯≥σn are the singular values of the adjacency matrix of D. In this paper we would like to prese...

Let G be a graph with the vertex set {v1,…,vn}. The Seidel matrix of G is an n×n matrix whose diagonal entries are zero, ij-th entry is −1 if vi and vj are adjacent and otherwise is 1. The Seidel energy of G, denoted by E(S(G)), is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix of G. Haemers conjectured that the Se...

A signed graph Gσ is an ordered pair (V(G),E(G)), where V(G) and E(G) are the set of vertices and edges of G, respectively, along with a map σ that signs every edge of G with +1 or −1. An eigenvalue of the associated adjacency matrix of Gσ, denoted by A(Gσ), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to th...

A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set {v1,…,vn}, is the matrix H=[hij]n×n, where hij=−hji=i if there is a directed edge from vi to vj, hij=1 if there exists an undirected edge between vi and vj, and hij=0 otherwise. The Hermitian spectrum of a mixed...

Let G be a graph and F : V ( G ) → 2 N be a mapping. The graph G is said to be F‐ avoiding if there exists an orientation O of G such that d O + ( v ) ∉ F ( v ) for every v ∈ V ( G ), where d O + ( v ) denotes the out‐degree of v in the directed graph G with respect to O. In this paper it is shown that if G is bipartite and ∣ F ( v ) ∣ ≤ d G ( v )...

A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set $ \{v_{1}, \ldots , v_{n}\} $, is the matrix $ H=[h_{ij}]_{n \times n} $, where $ h_{ij}=-h_{ji}=i $ if there is a directed edge from $ v_{i} $ to $ v_{j} $, $ h_{ij}=1 $ if there exists an undirected edge betwe...

A weighted graph $G^{\omega}$ consists of a simple graph $G$ with a weight $\omega$, which is a mapping,$\omega$: $E(G)\rightarrow\mathbb{Z}\backslash\{0\}$. A signed graph is a graph whose edges are labeled with $-1$ or $1$. In this paper, we characterize graphs which have a sign such that their signed adjacency matrix has full rank, and graphs wh...

In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumption...

In this paper, we study graphs whose matching polynomials have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs. We show that apart from K7(set minus)(E(C3)∪E(C4)) there is no connected k-regular matching integral graph if k≥2. It is als...

It was conjectured by Hoffmann-Ostenhof that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a family of cycles. In this paper, we show that this conjecture holds for traceable cubic graphs.

Let $G$ be a graph. A subgraph $H$ of $G$ is called a Sachs subgraph if each component of $H$ is either a copy of $K_2$ or a $2$-regular subgraph of $G$. The order of the largest Sachs subgraph of $G$ is called the perrank of $G$. A graph $G$ of order $n$ has full perrank if ${\rm perrank}(G)=n.$ In this article we characterize the family of all gr...

Let $G$ be a group. The \emph{power graph} of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chr...

Let and denote the group of rth roots of unity, the set of all matrices over , the matrix with all entries 1 and the matrix with in the (1, 1)-entry and 1 elsewhere, respectively. Let . Wang proved that if and only if A can be obtained from by a finite sequence of elementary operations. In this paper, we generalize this result for matrices over by...

Let be a graph and be a set of non-negative integers. By an -degree free spanning forest of we mean a spanning forest of with no vertex degree in . In this paper we study the existence of -degree free spanning forests in graphs. We show that if is a graph with minimum degree at least 4, then there exists a -degree free spanning forest. Moreover, we...

An unoriented flow in a graph, is an assignment of real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. This is equivalent to a flow in a bidirected graph all of whose edges are extraverted. A nowhere-zero unoriented k-flow is an unoriented flow with values from the set {+/- 1, ... , (k - 1)}. I...

Let G be a graph of minimum degree k. R.P. Gupta proved the two following interesting results:
1) A bipartite graph G has a k-edge-coloring in which all k colors appear at each vertex.
2) If G is a simple graph with minimum degree k > 1, then G has a (k−1)-edge-coloring in which all (k-1) colors appear at each vertex.
Let t be a positive integer. I...

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total
dominating set if every vertex of $G$ is adjacent to at least one vertex of
$S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality
of a total dominating set of $G$. In this paper using a greedy algorithm we
provide an upper bound for $\gamma_{t}$($G$), whe...

Let $G$ be a graph of order $n$. For every $v\in V(G)$, let $E_G(v)$ denote
the set of all edges incident with $v$. A signed $k$-submatching of $G$ is a
function $f:E(G)\longrightarrow \{-1,1\}$, satisfying $f(E_G(v))\leq 1$ for at
least $k$ vertices, where $f(S)=\sum_{e\in S}f(e)$, for each $ S\subseteq
E(G)$. The maximum of the value of $f(E(G))$...

Let G be a simple graph of order n. We mean by dominating set, a set S subset of V(G) such that every vertex of G is either in S or adjacent to a vertex in S. The domination polynomial of G is the polynomial Sigma(n)(i=1) d(G, i)x(i), where d(G, i) is the number of dominating sets of G of size i. Two graphs G and H are said to be D-equivalent, writ...

Let R be a ring with identity and M be a unitary left R-module. The complement of the intersection graph of submodules of M, denoted by Γ(M), is defined to be a graph whose vertices are in one-to-one correspondence with all nontrivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have zero i...

Let R be a commutative ring with nonzero identity. The Jacobson graph of R denoted by JR is a graph with the vertex set R\J(R), and two distinct vertices x, y ∈ V(JR) are adjacent if and only if 1-xy ∉ U(R), where U(R) is the set of all unit elements of R. Let ω (JR) denote the clique number of JR. It was conjectured that if R ≅{Πi=1n Ri is a commu...

For a set
${\mathcal{S}}$
of positive integers, a spanning subgraph F of a graph G is called an
${\mathcal{S}}$
-factor of G if
${\deg_F(x) \in \mathcal{S}}$
for all vertices x of G, where degF
(x) denotes the degree of x in F. We prove the following theorem on {a, b}-factors of regular graphs. Let r ≥ 5 be an odd integer and k be either an e...

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ ' (R) is a graph with the vertex set W * (R), where W * (R) is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a∉Rb and b∉Ra. In this paper, we show that if Γ ' (R) is a forest, then Γ ' (R) is a union...

Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Recently, it has been prov...

Let R be a commutative ring with unity and R +, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by (R) and G R , the Cayley graph Cay(R +, Z*(R)) and the unitary Cayley graph Cay(R +, U(R)), respectively. For an Artinian ring R, Akhtar et al. (2009) studied G R...

Let G and H be two graphs. A proper vertex coloring of G is called a dynamic coloring, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic coloring with k colors denoted by χ2(G). We denote the cartesian product of G and H by G□H. In this paper, we...

Let $G$ be a graph of order $n$. 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 $f(S) = \sum_{u\in S} f(u)$.
The minimum value of $ f(V(G))$, taken over all good functions is denoted by $
\gamma(G) $. A function $f:V(G)\rightarrow \{-1,...

Let [Inline formula] be a ring and [Inline formula] be the set of all non-trivial left ideals of [Inline formula]. The intersection graph of ideals of [Inline formula], denoted by [Inline formula], is a graph with the vertex set [Inline formula] and two distinct vertices [Inline formula] and [Inline formula] are adjacent if and only if [Inline form...

Let S⊆C⁎=C\{0}S⊆C⁎=C\{0} and A∈Mn(C)A∈Mn(C). The matrix A is called an S-GHMnGHMn if A∈Mn(S)A∈Mn(S) and AA⁎=Diag(λ1,…,λn)AA⁎=Diag(λ1,…,λn), for some positive numbers λi,i=1,…,nλi,i=1,…,n. In this paper we provide some necessary conditions on n for the existence of an S-GHMnGHMn over a finite set S. We conjecture that for every positive integer n, t...

A proper vertex coloring of a graph G is called a dynamic coloring if for every vertex ν with degree at least 2, the neighbors of ν receive at least two different colors. It was conjectured that if G is a regular graph, then χ2(G) - χ (G) ≤ 2. In this paper we prove that, apart from the cycles C4 and C5 and the complete bipartite graphs Kn,n, every...

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Let α(Γ'(R)) and γ(Γ'(R...

Let G = {g1,...,gn} be a finite abelian group. Consider the complete graph with the vertex set {g1.....,.....g n}. The G-coloring of Kn is a proper edge coloring in which the color of edge {gi,gj} gi g i + gj, 1 ≤ i < 3 ≤ n. We prove that in the G-coloring of the complete graph Kn, there exists a multicolored Hamilton path if G is not an elementary...

It is well-known that all starlike trees, i.e. trees with exactly one vertex of degree at least three, are determined by their Laplacian spectrum. A double starlike tree is a tree with exactly two vertices of degree at least three. In 2009, the following question was posed: Are all the double starlike trees determined by their Laplacian spectra? In...

A Roman dominating function on a graph G = (V(G), E(G)) is a labelling
${f : V(G)\rightarrow \{0,1,2\}}$
satisfying the condition that every vertex with label 0 has at least a neighbour with label 2. The Roman domination number
γ
R
(G) of G is the minimum of
${\sum_{v \in V(G)}{f(v)}}$
over all such functions. The Roman bondage number
b
R
(G) o...

Let R be a ring with unity and I(R)* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)* and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and so...

Let G be a simple graph of order n and size m. An edge covering of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. In this paper we introduce a new graph polynomial. The edge cover polynomial of G is the polynomial E(G,x)=@?"i"="1^me(G,i)x^i, where e(G,i) is the number of edge coverings of...

Suppose that R is a commutative ring with identity. Let A(R) be the set of all ideals of R with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)∗=A(R)∖{(0)} and two distinct vertices I and J are adjacent if and only if IJ=(0). In Behboodi and Rakeei (2011) [8], it was conjectured that f...

Let R be a ring and X⊆R be a non-empty set. The regular graph of X, Γ(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Γ(GLn(F)) finite? In...

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that
every pair of colors appears on at most one pair of adjacent vertices. The
harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors
needed for a harmonious coloring of $G$. We show that if $T$ is a forest of
order $n$ with maximum degree $\Delta(T)\geq \fra...

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-module M, denoted by G(M), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonze...

Let G be a simple graph and (G) denote the maximum degree of G. A harmonious colouring of G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. In this paper it is shown that n if T is a tree of order n and (T) , th...

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a
graph $\Gamma_G$ (called non-commuting graph of $G$) with $G$ as follows: take
$G\setminus Z(G)$ as the vertices of $\Gamma_G$ 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...

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 tha...

A colorful path in a graph G is a path with χ(G) vertices whose colors are different. A v-colorful path is such a path, starting from v. Let G≠C 7 be a connected graph with maximum degree Δ(G). We show that there exists a (Δ(G)+1)-coloring of G with a v-colorful path for every v∈V(G). We also prove that this result is true if one replaces (Δ(G)+1)...

Let R be a commutative ring and A(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)(0) and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilat...

Let G be a graph of order n with signless Laplacian eigenvalues q(1),...,q(n) and Laplacian eigenvalues mu(1),...,mu(n). It is proved that for any real number alpha with 0 < alpha <= 1 or 2 <= alpha < 3, the inequality q(1)(alpha) + ... + q(n)(alpha) >= mu(alpha)(1) + ... + mu(alpha)(n) holds, and for any real number beta with 1 < beta < 2, the ine...

A matrix A over a field F is said to be an AJT matrix if there exists a vector x over F such that both x and Ax have no zero component. The Alon–Jaeger–Tarsi (AJT) conjecture states that if F is a finite field, with |F|≥4, and A is an element of GL n (F) , then A is an AJT matrix. In this paper we prove that every nonzero matrix over a field F, wit...

Let G be a graph of order n such that \(\sum_{i=0}^{n}(-1)^{i}a_{i}\lambda^{n-i}\) and \(\sum_{i=0}^{n}(-1)^{i}b_{i}\lambda^{n-i}\) are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a
i
≥b
i
for i=0,1,…,n. As a consequence, we prove that for any α, 0<α≤1, if q
1,…,q
n
and μ
1,…,...

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=∑i=1nd(G,i)xi, where d(G,i) is the number of dominating sets of G of size i. A root of D(G,x) is called a domination root of G. We denote the set of distinct domination roots by Z(D(G,x)). Two graphs G and H are said to be D-equivalent, written as G∼H, if D(...

For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex
is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet’s Conjecture for bidirected graphs
are e...

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ(kG), where G is a finite group, k is a field, and 0 ≠ |G| k. Among other results it is shown that if...

Let $\gamma'_s(G)$ be the signed edge domination number of G. In 2006, Xu conjectured that: for any $2$-connected graph G of order $ n (n \geq 2),$ $\gamma'_s(G)\geq 1$. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer $m$, there exists an $m$-connected graph $G$ such that $ \gamma'_s(G...

For an integer ℓ≥2, a graph G is said to be a (0modℓ)-cycle graph if every cycle in G has length divisible by ℓ. So a graph is a (0mod2)-cycle graph if and only if it is bipartite. We prove the following results. (1) In contrast with bipartite graphs, whose list chromatic number can be arbitrary large, a (0modℓ)-cycle graph with ℓ≥3 has the list ch...

Let G be a simple graph of order n. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial I(G,x)=∑ k=0 n s k x k , where s k is the number of independent sets of G of size k and s 0 =1. It was proved that all roots of the independence polynomial of a claw-free graph are real. In...

A proper vertex k-coloring of a graph G is called dynamic, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a k-dynamic coloring is called the dynamic chromatic number of G and denoted by χ 2 (G). In this paper we study the dynamic chromatic number of grap...

Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R, are adjacent if and only if x+y∈Z(R), where Z(R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. Let R be a c...

Let F be a field, char(F)≠2, and S⊆GLn(F), where n is a positive integer. In this paper we show that if for every distinct elements x,y∈S, x+y is singular, then S is finite. We conjecture that this result is true if one replaces field with a division ring.

Let G be a simple graph of order n and size m which is not a tree. If ℓ≥3 is a natural number and the length of every cycle of G is divisible by ℓ, then m≤ℓ ℓ-2(n-2), and the equality holds if and only if the following hold: (i) ℓ is odd and G is a cycle of order f or (ii) ℓ is even and G is a generalized θ-graph with paths of length 2. It is shown...

Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination polynomial of $G$ is the polynomial $D(G,x)=\sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. In this paper we sho...

A proper vertex coloring of a graph G is called a dynamic coloring if for every vertex v of degree at least 2, the neighbors of v receive at least two different colors. Assume that ch2(G) is the minimum number k such that for every list assignment of size k to each vertex of G, there is a dynamic coloring of G such that every vertex is colored with...

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}, where d(G,i) is the number of dominating sets of G of size i, and \gamma(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the...

The energy of a graph/matrix is the sum of the absolute values of its eigenvalues. We investigate the result of duplicating/removing an edge to the energy of a graph. We also deal with the problem that which graphs G have the property that if the edges of G are covered by some subgraphs, then the energy of G does not exceed the sum of the subgraphs...

We say that two graphs G1 and G2 with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph Kn,n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices o...

Let G be a graph and χl(G) denote the list chromatic number of G. In this paper we prove that for every graph G for which the length of each cycle is divisible by l (l≥3), χl(G)≤3.

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(Mn(F)), where F is a field a...

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved that E(G)>= 2(n-\chi(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>= 2ch(G) for all graphs G except for those in a few specified families, where \bar{G}, \chi(G), and ch(G) are the complement, the chromat...

The zero-divisor graph of a ring R is defined as the directed graph Γ(R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x→y is an edge if and only if xy=0. Recently, it has been shown that for any finite ring R, Γ(R) has an even number of edges. Here we give a simple proof for this result. In t...

Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, n⩽rank(G)+rank(G¯)⩽2n. In this paper we characterize all graphs G such that rank(G)+rank(G¯)=n,n+1 or n + 2. Also for every integer n ⩾ 5 and any k, 0 ⩽ k ⩽ n, we construct a graph G of order n, such that rank(G)+rank(G¯)=n+k.

A multicolored tree is a tree whose edges have different colors. Brualdi and Hollingsworth [5] proved in any proper edge coloring of the complete graph K2n(n > 2) with 2n - 1 colors, there are two edge-disjoint multicolored spanning trees. In this paper we generalize this result showing that if (a1,…, ak) is a color distribution for the complete gr...

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph K is decomposable into commuting perfect matchings if and only if n is a 2-power. Also, it is shown that the complete graph Kn is decomposable into commuting Hamilton cy...

We study graphs whose adjacency matrices have determinant equal to 1 or −1, and characterize certain subclasses of these graphs. Graphs whose adjacency matrices are totally unimodular are also characterized. For bipartite graphs having a unique perfect matching, we provide a formula for the inverse of the corresponding adjacency matrix, and address...

In this paper we consider optimal edge colored complete graphs. We show that in any optimal edge coloring of the complete graph K n , there is a Hamil-ton cycle with at most √ 8n different colors. We also prove that in every proper edge coloring of the complete graph K n , there is a rainbow cycle with at least n/2−1 colors (A rainbow cycle is a cy...

The Silences of the Archives, the Reknown of the Story.
The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, befor...

Let G be a graph with a nonempty edge set, we denote the rank of the adjacency matrix of G and term rank of G, by rk(G) and Rk(G), respectively. van Nuffelen conjectured that for any graph G, χ(G)⩽rk(G). The first counterexample to this conjecture was obtained by Alon and Seymour. In 2002, Fishkind and Kotlov proved that for any graph G, χ(G)⩽Rk(G)...

Let G be a graph and for any natural number r, χs′(G,r) denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623–626]...

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ⩾ 3. In this paper we investigate the diameters of Γ(Mn(D)) and determine the diameters of some induced subgraphs of Γ(Mn(D)), such as the...

Let G be a graph with n vertices and m edges and assume that f:V(G)→N is a function with ∑v∈V(G)f(v)=m+n. We show that, if we can assign to any vertex v of G a list Lv of size f(v) such that G has a unique vertex coloring with these lists, then G is f-choosable. This implies that, if ∑v∈V(G)f(v)>m+n, then there is no list assignment L such that |Lv...

Let D be a division ring with center F and n⩾1 a natural number. For S⊆Mn(D) the commuting graph of S, denoted by Γ(S), is the graph with vertex set S⧹Z(S) such that distinct vertices a and b are adjacent if and only if ab=ba. In this paper we prove that if n>2 and A,N,I,T are the sets of all non-invertible, nilpotent, idempotent matrices, and invo...

A subgraph in an edge-colored graph is multicolored if all its edges receive distinct colors. In this paper, we prove that a complete graph on 2m (m �= 2) vertices K2m can be properly edge-colored with 2m − 1 colors in such a way that the edges of K2m can be partitioned into m multicolored isomorphic spanning trees.

Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph ΓG (called non-commuting graph of G) with G as follows: Take G\Z(G) as the vertices of ΓG and join two distinct vertices x and y, whenever xy≠yx. We want to explore how the graph theoretical properties of ΓG can effect on the group theoretical properties of G. We conjec...

In this paper we study the structure of some special bases for the null space of the incidence matrix of a graph. Recently it was shown that if G is a graph with no cut vertex, then G has a {−1, 0, 1}-basis. We generalize this result showing that the statement remains valid for every graph with no cut edge. For the null space of any bipartite graph...

We show that for any forest there exists a labelling of the vertices for which the row-reduced echelon form of its adjacency matrix is a {-1,0,1}-matrix. This result clearly provides an affirmative answer to the conjecture: The null space of the adjacency matrix of every forest has a {-1,0,1}-basis.

In a manner analogous to the commutative case, the zero-divisor graph of a non-commutative ring R can be defined as the directed graph Γ(R)Γ(R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x→yx→y is an edge if and only if xy=0xy=0. We investigate the interplay between the ring-theoretic prope...

Let R be a ring and f(x1,…,xn) be a polynomial in noncommutative indeterminates x1,…,xn with coefficients from Z and zero constant. The ring R is said to be an f-ring if f(r1,…,rn)=0 for all r1,…,rn of R and a virtually f-ring if for every n infinite subsets X1,…,Xn (not necessarily distinct) of R, there exist n elements r1∈X1,…,rn∈Xn such that f(r...