arXiv subjects

Irena M. Jovanović

Publications and source records attributed to Irena M. Jovanović.

3 recordsLinked to original sources

arXiv2026

Two conjectures on the cospectrality of complete tripartite graphs are false

For a graph G of order n with adjacency eigenvalues lambda_1(G) >= ... >= lambda_n(G), the cospectrality of G within the set G of all graphs of order n is the smallest spectral distance from G to another graph of G. Abdollahi and Zakeri conjectured that for every n >= 4 the cospectrality of the complete tripartite graph K_{n,n,n}, both in the Manhattan and in the Euclidean spectral distance, is attained precisely at K_{n-1,n,n+1}. We show that both conjectures are false for every n >= 4, that is, in the whole range in which they are stated. The counterexample is the graph H_n obtained from K_{n-1,n,n} by adding a vertex adjacent to the part of cardinality n - 1, and it is the natural generalization of the graph which the authors themselves exhibited for n = 3 and considered to be exceptional. We determine the spectrum of H_n in a closed form, and we show that the two spectral distances tend to 1 and to 1/2 respectively, instead of to (2*sqrt(3))/3 and to 2/3 as conjectured. On the other hand, we show that in the Euclidean distance the conjecture becomes asymptotically true when the competing graphs are restricted to complete multipartite graphs, so that it fails because a graph of rank four beats the conjectured minimizer.

math.CO↗

arXiv2026

Resolutions of two conjectures on the spectral diameter

Let $λ_1(G) \geq \dots \geq λ_n(G)$ be the adjacency spectrum of a graph $G$ on $n$ vertices. The spectral distance $σ(G,H)$ between $n$-vertex graphs $G$ and $H$ is the Manhattan distance between their spectra, i.e. $σ(G,H) = \sum_{i=1}^n |λ_i(G) - λ_i(H)|$. Given a set $\mathcal{G}$ of pairwise non-isomorphic graphs of order $n$, the spectral diameter of $\mathcal{G}$ is defined as $\mathrm{sdiam}(\mathcal{G}) = \max\{\mathrm{secc}_{\mathcal{G}}(G) : G \in \mathcal{G}\}$, where $\mathrm{secc}_{\mathcal{G}}(G) = \max\{σ(G,H) : H \in \mathcal{G}\}$ is the spectral eccentricity of $G \in \mathcal{G}$. Among six conjectures on spectral distances posed by Z. Stanić in 2012, two conjectures related to the spectral diameter of certain graph classes remained open. One of them concerns the spectral diameter of the set $\mathcal{B}_n$ of all connected bipartite graphs of order $n$, while the other, of the set $\mathcal{T}_n$ of all trees of order $n$. More precisely, Stanić conjectured that $\mathrm{sdiam}(\mathcal{T}_n) = σ(P_n, K_{1,n-1})$, where $P_n$ is the path graph, while $K_{1,n-1}$ is the star, and that $\mathrm{sdiam}(\mathcal{B}_n) = \mathrm{secc}_{\mathcal{B}_n}(K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor})$, where $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor}$ is the complete bipartite graph. In this paper, both of these conjectures are disproved.

math.CO↗

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.

math.CO↗