arXiv subjects

Saieed Akbari

Publications and source records attributed to Saieed Akbari.

At least 19 recordsLinked to original sources

arXiv2026

The switching conjecture for main eigenvalues is asymptotically true

An eigenvalue of a signed graph is called \emph{main} if there exists a corresponding eigenvector non-orthogonal to the all-ones vector. An important result of O'Rourke and Touri (2016) states that almost all (unsigned) graphs have all main eigenvalues. Akbari, França, Ghasemian, Javarsineh, and de Lima (2021) considered main eigenvalues of signed graphs and conjectured that for any unsigned connected graph $G \notin\{ K_2, K_4 - e\}$, there is a switching $\mathbf{s}$ such that all eigenvalues of the signed graph $G^{\mathbf{s}}$ are main. We prove two incomparable asymptotic versions of this conjecture. We show that for any graph $G$ of order $n$, there exists a switching $\mathbf{s}\in\{\pm1\}^n$ such that $G^{\mathbf{s}}$ has $n - O\!\left(\frac{n}{(\log n)^{1/4}}\right)$ main eigenvalues counted with multiplicity. Using a similar proof strategy, we also show that if $G$ has $d$ distinct eigenvalues, then there exists a switching $\mathbf{s}\in\{\pm1\}^n$ such that $G^{\mathbf{s}}$ has $d - O\!\left(\frac{d}{(\log d)^{1/4}}\right)$ main eigenvalues.

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arXiv2026

Spectrally symmetric orientations of graphs

The Hermitian adjacency matrices of digraphs based on the sixth root of unity were introduced in [B. Mohar, A new kind of Hermitian matrices for digraphs, Linear Alg. Appl. (2020)]. They appear to be the most natural choice for the spectral theory of digraphs. Undirected graphs have adjacency spectrum symmetric about 0 if and only if they are bipartite. The situation is more complex for the Hermitian spectra of digraphs. In this paper we study non-bipartite oriented graphs with symmetric Hermitian spectra. Our main result concerns the extremal problem of maximizing the density of spectrally symmetric oriented graphs. The maximum possible density is shown to be between 31/36} and 10/11. Furthermore, we give a necessary condition for an oriented graph to be spectrally symmetric based on the adjacency spectrum of the underlying graph. This allows us to show that line graphs of sufficiently dense graphs do not admit spectrally symmetric orientations. We also show how to construct infinite families of spectrally symmetric graphs using 1-sums.

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arXiv2026

Bounds for the Vertex Chromatic Number of Connected Triangle-Free Graphs

It was recently shown that every connected graph of order $n \geq 5$ and size $m$ satisfies $χ(G) \leq \left\lceil \frac{m}{\sqrt{n}} \right\rceil$, and it was asked whether the stronger inequality $χ(G) \leq \left\lceil \frac{m}{2\sqrt{n}} \right\rceil + 1$ holds for every connected triangle-free graph. In this paper, we answer this question in the affirmative. In fact, we prove that every connected triangle-free graph $G$ with $G \not\cong C_5$ satisfies $χ(G) \leq \left\lceil \frac{m}{\sqrt{5.5n}} \right\rceil + 1$, where the constant $\sqrt{5.5}$ cannot be replaced by any constant greater than or equal to $\sqrt{6}$, and the equality holds for the Grötzsch graph and for every odd cycle of length between $7$ and $21$.

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arXiv2026

An infinite family of trees with irreducible characteristic polynomials

Consider the tree obtained by attaching a leaf to the third vertex of a path with n-1 vertices. In this note, we prove that the characteristic polynomial of this tree is irreducible when n belongs to certain arithmetic progressions modulo 30. As a result there are infinitely many pairwise non-isomorphic trees with an irreducible the characteristic polynomial. Our proof combines several number-theoretic arguments with a result of Gross, Hironaka, and McMullen [GHM09] concerning the cyclotomic factors of the Coxeter polynomials associated with the diagrams En. This resolves affirmatively a conjecture of Akbari, Kumar, Mohar and Pragada.

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arXiv2026

On the Spectra of Digraph Laplacians

We present several Laplacian-type matrices associated with a loopless digraph $D$: the out-/in-degree Laplacians $\mathcal L_{\mathrm{out}},\mathcal L_{\mathrm{in}}$, the incidence Laplacian $\mathcal L_{\mathrm{inc}}=BB^{\mathsf T}$, and the symmetrized and skew-symmetrized variants $\mathcal S_{\mathrm{out}},\mathcal K_{\mathrm{out}}$. We show that $\mathcal L_{\mathrm{inc}}(D)$ coincides with the Laplacian of the underlying undirected multigraph, and we derive spectral and characteristic-polynomial relations under arc reversal and complementation (including a simplification for Eulerian digraphs for $\mathcal S_{\mathrm{out}}$). We demonstrate that the spectral radius of $\mathcal L_{\mathrm{out}}$ is bounded above by the order of the digraph and give a characterization in the equality case. We further obtain explicit formulas for joins and line digraphs, giving a general determinantal identity relating the out-degree Laplacian characteristic polynomials of a regular digraph and its line digraph.

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arXiv2026

Distribution of signless Laplacian eigenvalues and degree sequence

Let $G$ be a graph of order $n$ with degree sequence $d_1 \geq \cdots \geq d_{n}$. Let $m_{G}I$ be the number of signless Laplacian eigenvalues in an interval $I$. In this paper, we characterize the distribution of the signless Laplacian eigenvalues in terms of the degree sequence of a graph within specific subintervals of $[0, \, 2n-2].$ We determine all graphs $G$ such that $m_{G}[d_n, 2n-2] \leq 2, \; m_{G}[d_{n-1}, 2n-2] = 1, \; m_{G}[0, d_1] \le 2.$ We also prove that there is no graph such that $m_{G}[0, d_3]=1$. In addition, we obtain all disconnected graphs such that $m_{G}[0, d_1] = 3$. Finally, we propose two open problems for future research.

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arXiv2025

On Prime Matrix Product Factorizations

A graph $G$ factors into graphs $H$ and $K$ via a matrix product if $A = BC$, where $A$, $B$, and $C$ are the adjacency matrices of $G$, $H$, and $K$, respectively. The graph $G$ is prime if, in every such factorization, one of the factors is a perfect matching that is, it corresponds to a permutation matrix. We characterize all prime graphs, then using this result we classify all factorable forests, answering a question of Akbari et al. [\emph{Linear Algebra and its Applications} (2025)]. We prove that every torus is factorable, and we characterize all possible factorizations of grids, addressing two questions posed by Maghsoudi et al. [\emph{Journal of Algebraic Combinatorics} (2025)].

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arXiv2025

A new conjecture on the inertia of graphs

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torgašev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph $G$, its line graph $L(G)$ satisfies $n^+(L(G)) \le n^-(L(G)) + 1$, and obtain partial results.

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arXiv2025

Hermitian adjacency matrices with at most three distinct eigenvalues

We study oriented graphs whose Hermitian adjacency matrices of the second kind have few eigenvalues. We give a complete characterization of the oriented graphs with two distinct eigenvalues, showing that there are only four such graphs. We extend this result to mixed graphs. We show that there are infinitely many regular tournaments with three distinct eigenvalues. We extend our main results to Hermitian adjacency matrices defined over other roots of unity.

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arXiv2025

Tight Bounds for Cycle-Edge Decompositions and Covers

An old conjecture of Erd{ő}s and Gallai states that every $n$ vertex graph can be decomposed, that is $E(G)$ can be partitioned, into $O(n)$ cycles and edges. The covering version of this conjecture was proven by Pyber in 1985, where it was shown that all graphs can be covered by $n-1$ cycles and edges. The best upper bound on the number of cycles and edges required to decompose any graph is $O(n\log^*(n))$, which was recently shown by Buci{ć} and Montgomery in 2023. Here $\log^*(n)$ denotes the iterated logarithm function. Meanwhile, a construction of Erdős demonstrate that there exists graphs which require $(\frac{3}{2}-o(1))n$ cycles and edges to be decomposed. We prove all graphs with maximum degree at most $4$ can be decomposed into $n-1$ or fewer cycles and edges. We also show that every $n$ vertex claw-free graph can be decomposed into $n-1$ or fewer $2$-regular subgraphs and edges. Finally, we prove that every graph $G$ containing a cycle can be covered by $n-2$ or fewer cycles and edges. This improves Pyber's covering theorem by proving that $n-1$ cycles and edges are required only for trees.

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arXiv2025

Vertex Partitioning and $p$-Energy of Graphs

For a Hermitian matrix $A$ of order $n$ with eigenvalues $λ_1(A)\ge \cdots\ge λ_n(A)$, define \[ \mathcal{E}_p^+(A)=\sum_{λ_i > 0} λ_i^p(A), \quad \mathcal{E}_p^-(A)=\sum_{λ_i<0} |λ_i(A)|^p,\] to be the positive and the negative $p$-energy of $A$, respectively. In this note, first we show that if $A=[A_{ij}]_{i,j=1}^k$, where $A_{ii}$ are square matrices, then \[ \mathcal{E}_p^+(A)\geq \sum_{i=1}^{k} \mathcal{E}_p^+(A_{ii}), \quad \mathcal{E}_p^-(A)\geq \sum_{i=1}^{k} \mathcal{E}_p^-(A_{ii}),\] for any real number $p\geq 1$. We then apply the previous inequality to establish lower bounds for $p$-energy of the adjacency matrix of graphs.

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arXiv2025

Refinement of a conjecture on positive square energy of graphs

Let $G$ be a simple graph of order $n$ with eigenvalues $λ_1(G)\geq \cdots \geq λ_n(G)$. Define \[s^+(G)=\sum_{λ_i >0} λ_i^2(G), \quad s^-(G)=\sum_{λ_i<0} λ_i^2(G).\] It was conjectured by Elphick, Farber, Goldberg and Wocjan that for every connected graph $G$ of order $n$, $s^+(G) \ge n-1.$ We verify this conjecture for graphs with domination number at most 2. We then strengthen the conjecture as follows: if $G$ is a connected graph of order $n$ and size $m \geq n+1$, then $s^+(G) \geq n$. We prove this conjecture for claw-free graphs and graphs with diameter 2.

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arXiv2025

On the $Δ$-edge stability number of graphs

The $Δ$-edge stability number ${\rm es}_Δ(G)$ of a graph $G$ is the minimum number of edges of $G$ whose removal results in a subgraph $H$ with $Δ(H) = Δ(G)-1$. Sets whose removal results in a subgraph with smaller maximum degree are called mitigating sets. It is proved that there always exists a mitigating set which induces a disjoint union of paths of order $2$ or $3$. Minimum mitigating sets which induce matchings are characterized. It is proved that to obtain an upper bound of the form ${\rm es}_Δ(G) \leq c |V(G)|$ for an arbitrary graph $G$ of given maximum degree $Δ$, where $c$ is a given constant, it suffices to prove the bound for $Δ$-regular graphs. Sharp upper bounds of this form are derived for regular graphs. It is proved that if $Δ(G) \geq\frac{|V(G)|-2}{3}$ or the induced subgraph on maximum degree vertices has a $Δ(G)$-edge coloring, then ${\rm es}_Δ(G) \le \lceil |V(G)|/2\rceil$.

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arXiv2024

A Linear Lower Bound for the Square Energy of Graphs

Let $G$ be a graph of order $n$ with eigenvalues $λ_1 \geq \cdots \geqλ_n$. Let \[s^+(G)=\sum_{λ_i>0} λ_i^2, \qquad s^-(G)=\sum_{λ_i<0} λ_i^2.\] The smaller value, $s(G)=\min\{s^+(G), s^-(G)\}$ is called the \emph{square energy} of $G$. In 2016, Elphick, Farber, Goldberg and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\geq n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1, \ldots, H_k$ be disjoint vertex-induced subgraphs of $G$. In this note, we prove that \[s^+(G)\geq\sum_{i=1}^{k} s^+(H_i) \quad \text{ and } \quad s^-(G)\geq\sum_{i=1}^{k} s^-(H_i),\] which implies that $s(G)\geq \frac{3n}{4}$ for every connected graph $G$ of order $n\ge 4$.

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arXiv2024

Lower bounds for the Randić index in terms of matching number

We investigate how small the Randić index of a graph can be in terms of its matching number, and prove several results. We give best-possible linear bounds for graphs of small excess and for subcubic graphs; in the former case the size of excess we permit is qualitatively the best possible. We show that a linear bound holds for any sparse hereditary graph class (such as planar graphs). In general, however, we show that it can be much smaller than linear. We determine the asymptotic growth rate of the minimum Randić index for graphs with a near perfect matching, and conjecture that the same bounds hold for all graphs.

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arXiv2024

Spectral Methods for Matrix Product Factorization

A graph $G$ is factored into graphs $H$ and $K$ via a matrix product if there exist adjacency matrices $A$, $B$, and $C$ of $G$, $H$, and $K$, respectively, such that $A = BC$. In this paper, we study the spectral aspects of the matrix product of graphs, including regularity, bipartiteness, and connectivity. We show that if a graph $G$ is factored into a connected graph $H$ and a graph $K$ with no isolated vertices, then certain properties hold. If $H$ is non-bipartite, then $G$ is connected. If $H$ is bipartite and $G$ is not connected, then $K$ is a regular bipartite graph, and consequently, $n$ is even. Furthermore, we show that trees are not factorizable, which answers a question posed by Maghsoudi et al.

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arXiv2024

Line graphs and Nordhaus-Gaddum-type bounds for self-loop graphs

Let $G_S$ be the graph obtained by attaching a self-loop at every vertex in $S \subseteq V(G)$ of a simple graph $G$ of order $n.$ In this paper, we explore several new results related to the line graph $L(G_S)$ of $G_S.$ Particularly, we show that every eigenvalue of $L(G_S)$ must be at least $-2,$ and relate the characteristic polynomial of the line graph $L(G)$ of $G$ with the characteristic polynomial of the line graph $L(\widehat{G})$ of a self-loop graph $\widehat{G}$, which is obtained by attaching a self-loop at each vertex of $G$. Then, we provide some new bounds for the eigenvalues and energy of $G_S.$ As one of the consequences, we obtain that the energy of a connected regular complete multipartite graph is not greater than the energy of the corresponding self-loop graph. Lastly, we establish a lower bound of the spectral radius in terms of the first Zagreb index $M_1(G)$ and the minimum degree $δ(G),$ as well as proving two Nordhaus-Gaddum-type bounds for the spectral radius and the energy of $G_S,$ respectively.

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arXiv2023

Tight Bounds on the Chromatic Edge Stability Index of Graphs

The chromatic edge stability index $\mathrm{es}_{χ'}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on $\mathrm{es}_{χ'}(G)$ in terms of the number of vertices of degree $Δ(G)$ (if $G$ is Class 2), and the numbers of vertices of degree $Δ(G)$ and ${Δ(G)-1}$ (if $G$ is Class 1). If $G$ is bipartite we give an exact expression for $\mathrm{es}_{χ'}(G)$ involving the maximum size of a matching in the subgraph induced by vertices of degree $Δ(G)$. Finally, we consider whether a minimum mitigating set, that is a set of size $\mathrm{es}_{χ'}(G)$ whose removal reduces the chromatic index, has the property that every edge meets a vertex of degree at least $Δ(G)-1$; we prove that this is true for some minimum mitigating set of $G$, but not necessarily for every minimum mitigating set of $G$.

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