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Zoran Stanić

Publications and source records attributed to Zoran Stanić.

9 recordsLinked to original sources
arXiv2026

Helmholzian Spectra of Graphs: Novel Properties

Let $\grad$, $\curl$, and $\dv$ be the graph-theoretic analogues of the gradient, curl, and divergence operators from multivariate calculus. The graph Laplacian $-\dv \grad$ gives rise to the celebrated Laplacian matrix, while the matrix representation of the graph Helmholtzian $\grad \grad^* + \curl^* \curl$ is called the Helmholtzian matrix. In this paper, we present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. We then investigate the spectral properties of this matrix. Our main results are as follows: (i) a classification of graphs having exactly two distinct Helmholtzian eigenvalues; (ii) the nullity of the Helmholtzian matrix; and (iii) a combinatorial interpretation of the coefficients of the Helmholtzian polynomial. Furthermore, we determine the Helmholtzian spectrum for certain graph products and characterize Helmholtzian integral graphs, as well as derive bounds for the smallest Helmholtzian eigenvalue. Meanwhile, we pose some open problems for future research.

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arXiv2026

Helmholzian spectra of graphs: basic properties

The Helmholtzian matrix of a graph $G=(V(G),E(G))$ is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) [S. Li, L. Lu, J.F. Wang, A graph discretization of vector Laplacian, 379 (2026) 446--460]. Motivated by the applications of graph Helmholtzian in simplicial networks, we will investiagte its basic spectral properties. As the first graph matrix indexed by edge set, we find that Helmholtzian matrix is positive semi-definite and its non-negativity correlates with the odd cycles in $G$ and the orientation on $E(G)$, while its irreducibility relates to the signed graphs with loops. We show that the eigenvalues of Helmholtzian matrix are independent of the orientation and further investigate the eigenvalue interlacing under edge additions. One of striking findings is that the non-zero eigenvalues of the Laplacian matrix are those of Helmholtzian matrix of every graph. All these discoveries reveal that the Helmholtzian spectrum of $G$ balances and bridges the oriented graphs, weighted graphs and signed graphs as well as their adjacency or Laplacian spectra.

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arXiv2025

Signless Laplacian spectral analysis of a class of graph joins

A graph is said to be determined by its signless Laplacian spectrum (abbreviated as DQS) if no other non-isomorphic graph shares the same signless Laplacian spectrum. In this paper, we establish the following results: (1). Every graph of the form $K_1 \vee (C_s \cup qK_2)$, where $q \ge 0$, $s \ge 3$, and the number of vertices is at least $16$, is DQS; (2). Every graph of the form $K_1 \vee (C_{s_1} \cup C_{s_2} \cup \cdots \cup C_{s_t} \cup qK_2)$, where $t \ge 2$, $q \ge 0$, $s_i \ge 3$, and the number of vertices is at least $52$, is DQS. Here, $K_n$ and $C_n$ denote the complete graph and the cycle of order $n$, respectively, while $\cup$ and $\vee$ represent the disjoint union and the join of graphs. Moreover, the signless Laplacian spectrum of the graphs under consideration is computed explicitly.

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arXiv2025

Determining some graph joins by the signless Laplacian spectrum

A graph is determined by its signless Laplacian spectrum if there is no other non-isomorphic graph sharing the same signless Laplacian spectrum. Let $C_l$, $P_l$, $K_l$ and $K_{s,l-s}$ be the cycle, the path, the complete graph and the complete bipartite graph with $l$ vertices, respectively. We prove that $$G\cong K_1\vee (C_{l_1}\cup C_{l_2}\cup\cdots \cup C_{l_t}\cup sK_1),$$ with $s\ge 0, t\ge 1, n\geq 22$, is determined by the signless Laplacian spectrum if and only if either $s=0$ or $s\ge 1$ and $l_i\ne 3$ holds for all $1\leq i\leq t$, where $n$ is the order of $G$, and $\cup$ and $\vee$ stand for the disjoint union and the join of two graphs, respectively. Moreover, for $s\ge 1$ and $l_t=3$, $K_1\vee (K_{1,3}\cup C_{l_1}\cup C_{l_2}\cup\cdots \cup C_{l_{t-1}}\cup (s-1)K_1)$ is fixed as a graph sharing the signless Laplacian spectrum with $G$. This contribution extends some recently published results.

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arXiv2021

The main vertices of a star set and related graph parameters

A vertex $v \in V(G)$ is called $λ$-main if it belongs to a star set $X \subset V(G)$ of the eigenvalue $λ$ of a graph $G$ and this eigenvalue is main for the graph obtained from $G$ by deleting all the vertices in $X \setminus \{v\}$; otherwise, $v$ is $λ$-non-main. Some results concerning main and non-main vertices of an eigenvalue are deduced. For a main eigenvalue $λ$ of a graph $G$, we introduce the minimum and maximum number of $λ$-main vertices in some $λ$-star set of $G$ as new graph invariant parameters. The determination of these parameters is formulated as a combinatorial optimization problem based on a simplex-like approach. Using these and some related parameters we develop new spectral tools that can be used in the research of the isomorphism problem. Examples of graphs for which the maximum number of $λ$-main vertices coincides with the cardinality of a $λ$-star set are provided.

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arXiv2021

Signed $(0,2)$-graphs with few eigenvalues and a symmetric spectrum

We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two distinct eigenvalues $\pm λ$. Next, we consider to what extent induced subgraphs of signed graph with two distinct eigenvalues $\pm λ$ are determined by their spectra. Lastly, we classify signed $(0,2)$-graphs that have a symmetric spectrum with three distinct eigenvalues and give a partial classification for those with four distinct eigenvalues.

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arXiv2021

Total graph of a signed graph

The total graph is built by joining the graph to its line graph by means of the incidences. We introduce a similar construction for signed graphs. Under two similar definitions of the line signed graph, we define the corresponding total signed graph and we show that it is stable under switching. We consider balance, the frustration index and frustration number, and the largest eigenvalue. In the regular case we compute the spectrum of the adjacency matrix of the total graph and the spectra of certain compositions, and we determine some with exactly two main eigenvalues.

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arXiv2013

Probability and strategy in a variation of a blackjack game

The subject of this paper is a variation of a blackjack game, mainly popular in some parts of Europe where it is known as einz (in German slang: one). We describe the rules of this game, indicate its main characteristics, give some probabilities, suggest the strategy, and consider some typical situations.

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