arXiv subjects

Ebrahim Ghorbani

Publications and source records attributed to Ebrahim Ghorbani.

32 records · Page 2Linked to original sources

arXiv2018

Spectral properties of cographs and $P_5$-free graphs

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph $G$ is a cograph if and only if no induced subgraph of $G$ has an eigenvalue in the interval $(-1,0)$. It is also shown that the multiplicity of any eigenvalue of a cograph $G$ does not exceed the sum of multiplicities of $0$ and $-1$ as eigenvalues of $G$. We introduce a partial order on the vertex set of graphs $G$ in terms of inclusions among the open and closed neighborhoods of vertices, and conjecture that the multiplicity of any eigenvalue of a cograph $G$ except for $0,-1$ does not exceed the maximum size of an antichain with respect to that partial order. In two extreme cases (in particular for threshold graphs), the conjecture is shown to be true. Finally, we give a simple proof for the result that bipartite $P_5$-free graphs have no eigenvalue in the intervals $(-1/2,0)$ and $(0,1/2)$.

math.CO↗

arXiv2018

Cographs: Eigenvalues and Dilworth Number

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains with respect to the vicinal preorder required to cover the vertex set of a graph $G$ is called the Dilworth number of $G$. We prove that for any cograph $G$, the multiplicity of any eigenvalue $λ\ne0,-1$, does not exceed the Dilworth number of $G$ and show that this bound is tight. G. F. Royle [The rank of a cograph, Electron. J. Combin. 10 (2003), Note 11] proved that if a cograph $G$ has no pair of vertices with the same neighborhood, then $G$ has no 0 eigenvalue, and asked if beside cographs, there are any other natural classes of graphs for which this property holds. We give a partial answer to this question by showing that an $H$-free family of graphs has this property if and only if it is a subclass of the family of cographs. A similar result is also shown to hold for the $-1$ eigenvalue.

math.CO↗

arXiv2017

Some spectral properties of chain graphs

A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. Alazemi, Andelić and Simić conjectured that no chain graph shares a non-zero (adjacency) eigenvalue with its vertex-deleted subgraphs. We disprove this conjecture. However, we show that the assertion holds for subgraphs obtained by deleting vertices of maximum degrees in either of color classes. We also give a simple proof for the fact that chain graphs have no eigenvalue in the interval $(0,1/2)$.

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arXiv2016

Proof of a conjecture on `plateaux' phenomenon of graph Laplacian eigenvalues

Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both Laplacian and signless Laplacian eigenvalues from which the conjecture follows.

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arXiv2016

Nontrivial nuciferous graphs exist

A nuciferous graph is a simple graph with a non-singular $0$-$1$ adjacency matrix $A$ such that all the diagonal entries of $A^{-1}$ are zero and all the off-diagonal entries of $A^{-1}$ are non-zero. Sciriha et al. conjectured that except $K_2$, no nuciferous graph exists. We disprove this conjecture. Moreover, we conjecture that there infinitely many nuciferous Cayley graphs.

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arXiv2014

Spanning trees and even integer eigenvalues of graphs

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $τ(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $τ(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $λ\equiv2\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $λ\equiv0\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs. We also show that if $τ(G)=2^ts$ with $s$ odd, then the multiplicity of any even integer eigenvalue of $Q(G)$ is at most $t+1$. Among other things, we prove that if $L(G)$ or $Q(G)$ has an even integer eigenvalue of multiplicity at least $2$, then $τ(G)$ is divisible by $4$. As a very special case of this result, a conjecture by Zhou et al. [On the nullity of connected graphs with least eigenvalue at least $-2$, Appl. Anal. Discrete Math. 7 (2013), 250--261] on the nullity of adjacency matrices of the line graphs of unicyclic graphs follows.

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arXiv2013

On the eigenvalues of certain Cayley graphs and arrangement graphs

In this paper, we show that the eigenvalues of certain classes of Cayley graphs are integers. The (n,k,r)-arrangement graph A(n,k,r) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they differ in exactly r positions. We establish a relation between the eigenvalues of the arrangement graphs and the eigenvalues of certain Cayley graphs. As a result, the conjecture on integrality of eigenvalues of A(n,k,1) follows.

math.CO↗

arXiv2013

Cyclic decomposition of k-permutations and eigenvalues of the arrangement graphs

The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k-1 positions. We introduce a cyclic decomposition for k-permutations and show that this gives rise to a very fine equitable partition of A(n,k). This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of A(n,k). Consequently, we determine the eigenvalues of A(n,k) for small values of k. Finally, we show that any eigenvalue of the Johnson graph J(n,k) is an eigenvalue of A(n,k) and that -k is the smallest eigenvalue of A(n,k) with multiplicity O(n^k) for fixed k.

math.CO↗

arXiv2013

Inverses of triangular matrices and bipartite graphs

To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular triangular matrices A such that A and A^{-1} have the same zero-nonzero pattern are characterized. A combinatorial construction is given to construct outer inverses of the adjacency matrix of a weighted tree.

math.CO↗

arXiv2013

On eigenvalues of Seidel matrices and Haemers' conjecture

For a graph $G$, let $S(G)$ be the Seidel matrix of $G$ and $\te_1(G),...,\te_n(G)$ be the eigenvalues of $S(G)$. The Seidel energy of $G$ is defined as $|\te_1(G)|+...+|\te_n(G)|$. Willem Haemers conjectured that the Seidel energy of any graph with $n$ vertices is at least $2n-2$, the Seidel energy of the complete graph with $n$ vertices. Motivated by this conjecture, we prove that for any $\al$ with $0<\al<2$, $|\te_1(G)|^\al+...+|\te_n(G)|^\al\g (n-1)^\al+n-1$ if and only if $|{\rm det} S(G)|\g n-1$. This, in particular, implies the Haemers' conjecture for all graphs $G$ with $|{\rm det} S(G)|\g n-1$.

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arXiv2012

Graphs with few matching roots

We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.

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arXiv2011

Bipartite graphs with five eigenvalues and pseudo designs

A pseudo $(v,\, k,\, \la)$-design is a pair $(X, {\cal B})$ where $X$ is a $v$-set and ${\cal B}=\{B_1,...,B_{v-1}\}$ is a collection of $k$-subsets (blocks) of $X$ such that each two distinct $B_i, B_j$ intersect in $\la$ elements; and $0\le\la <k \le v-1$. We use the notion of pseudo designs to characterize graphs of order $n$ whose (adjacency) spectrum contains a zero and $\pmθ$ with multiplicity $(n-3)/2$ where $0<θ\le\sqrt{2}$. Meanwhile, partial results confirming a conjecture of O. Marrero on characterization of pseudo $(v,\, k,\, \la)$-designs are obtained.

math.CO↗

arXiv2009

Bounds for the Huckel energy of a graph

Let $G$ be a graph on $n$ vertices with $r := \lfloor n/2 \rfloor$ and let $λ_1 \geq...\geq λ_{n} $ be adjacency eigenvalues of $G$. Then the Hückel energy of $G$, HE($G$), is defined as $$\he(G) = {ll} 2\sum_{i=1}^{r} λ_i, & \hbox{if $n= 2r$;} 2\sum_{i=1}^{r} λ_i + λ_{r+1}, & \hbox{if $n= 2r+1$.} $$ The concept of Hückel energy was introduced by Coulson as it gives a good approximation for the $π$-electron energy of molecular graphs. We obtain two upper bounds and a lower bound for HE$(G)$. When $n$ is even, it is shown that equality holds in both upper bounds if and only if $G$ is a strongly regular graph with parameters $(n, k, λ, μ) = (4t^2 +4t +2, 2t^2 +3t +1, t^2 +2t, t^2 + 2t +1),$ for positive integer $t$. Furthermore, we will give an infinite family of these strongly regular graph whose construction was communicated by Willem Haemers to us. He attributes the construction to J.J. Seidel.

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arXiv2007

Choice Number and Energy of Graphs

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-χ(\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}, χ(G), and ch(G) are the complement, the chromatic number, and the choice number of G, respectively.

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