Riste Škrekovski

University of Ljubljana · Faculty of Mathematics and Physics

Let $W(G)$ be the Wiener index of a graph $G$. We say that a vertex $v \in V(G)$ is a \v{S}olt\'es vertex in $G$ if $W(G - v) = W(G)$, i.e. the Wiener index does not change if the vertex $v$ is removed. In 1991, \v{S}olt\'es posed the problem of identifying all connected graphs $G$ with the property that all vertices of $G$ are \v{S}olt\'es vertice...

The Wiener index is defined as the sum of distances between all unordered pairs of vertices in a graph. It is one of the most recognized and well-researched topological indices, which is on the other hand still a very active area of research. This work presents a natural continuation of the paper Mathematical aspects of Wiener index (Ars Math. Cont...

This paper proposes a new data structure, multiset-trie, that is designed for storing and efficiently processing a set of multisets. Moreover, multiset-trie can operate on a set of sets without efficiency loss. The multiset-trie structure is a search tree with properties similar to those of a trie. It implements all standard search tree operations...

We answer a question posed by Tibor Gallai in 1969 concerning criticality in Sperner's lemma, listed as Problem 9.14 in the collection of Jensen and Toft [Graph coloring problems, John Wiley & Sons, Inc., New York, 1995]. Sperner's lemma states that if a labelling of the vertices of a triangulation of the $d$-simplex $\Delta^d$ with labels $1, 2, \...

In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-colorings of graphs. A proper vertex coloring φ of a graph G is said to be odd if for each non-isolated vertex x∈V(G) there exists a color c such that φ−1(c)∩N(x) is odd-sized. We prove that every simple planar graph admits an odd 9-coloring,...

A proper vertex coloring φ of graph G is said to be odd if for each non-isolated vertex x∈V(G) there exists a color c such that φ−1(c)∩N(x) is odd-sized. The minimum number of colors in any odd coloring of G, denoted χo(G), is the odd chromatic number. Odd colorings were recently introduced in Petruševski and Škrekovski (0000). Here we discuss vari...

In a graph G, a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S. The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G. In Sedlar and Škrekovski (0000) we determined the vert...

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 85(2) (2023), 195–219. An assignment ϕ ∶ V (G) → Z is said to be an odd-sum coloring of graph G if no two adjacent vertices receive the same color (i.e. the coloring is proper) and for every vertex v ∈ V (G) the sum ∑ u∈N [v] ϕ(u) of all colors used in the closed neighborhood N [v] is odd. The minimum...

The vertex (resp. edge) metric dimension of a connected graph G, denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S⊆V(G) which distinguishes all pairs of vertices (resp. edges) in G. Bounds dim(G)≤L(G)+2c(G) and edim(G)≤L(G)+2c(G), where c(G) is the cyclomatic number in G and L(G) depends on the number of leaves in G, are...

Many RDF stores treat graphs as simple sets of vertices and edges without a conceptual schema. The statistics in the schema-less RDF stores are based on the cardinality of the keys representing the constants in triple patterns. In this paper, we explore the effects of storing knowledge graphs in an RDF store on the structure of the space of queries...

The vertex (respectively edge) metric dimension of a graph G is the size of a smallest vertex set in G, which distinguishes all pairs of vertices (respectively edges) in G, and it is denoted by dim(G) (respectively edim(G)). The upper bounds dim(G)≤2c(G)−1 and edim(G)≤2c(G)−1, where c(G) denotes the cyclomatic number of G, were established to hold...

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph...

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph...

The vertex (resp. edge) metric dimension of a connected graph G is the size of a smallest set S⊆V(G) which distinguishes all pairs of vertices (resp. edges) in G. In Sedlar and Škrekovski (2021) it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that...

The (independent) chromatic vertex stability ($\ivs(G)$) $\vs(G)$ is the minimum size of (independent) set $S\subseteq V(G)$ such that $\chi(G-S)=\chi(G)-1$. In this paper we construct infinitely many graphs $G$ with $\Delta(G)=4$, $\chi(G)=3$, $\ivs(G)=3$ and $\vs(G)=2$, which gives a partial negative answer to a problem posed in \cite{ABKM}.

The vertex (resp. edge) metric dimension of a graph G is the size of a smallest vertex set in G which distinguishes all pairs of vertices (resp. edges) in G and it is denoted by dim(G) (resp. edim(G)). The upper bounds dim(G) <= 2c(G) - 1 and edim(G) <= 2c(G)-1; where c(G) denotes the cyclomatic number of G, were established to hold for cacti witho...

A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., \textit{Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods}, arXiv preprint]...

A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum numb...

This paper is devoted to Wiener index of directed graphs, more precisely of directed grids. The grid $G_{m,n}$ is the Cartesian product $P_m\Box P_n$ of paths on $m$ and $n$ vertices, and in a particular case when $m=2$, it is a called the ladder graph $L_n$. Kraner \v{S}umenjak et al. proved that the maximum Wiener index of a digraph, which is obt...

A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of $G$, denoted $\chi_o(G)$, is the odd chromatic number. Odd colorings were recently introduced in [M.~Petru\v{s}e...

The line graph L(G) of a graph G is defined as a graph having vertex set identical with the set of edges of G and two vertices of L(G) are adjacent if and only if the corresponding edges are incident in G. Higher iteration L i(G) is obtained by repeatedly applying the line graph operation i times. Wiener index W(G) of a graph G is defined as the su...

In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-colorings of graphs. A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. We prove that every simple planar gr...

A locally irregular graph is a graph in which the end vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it allows a locally irregular edge coloring. The locally irregular chromatic in...

A locally irregular graph is a graph in which the end-vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it admits a locally irregular edge coloring. The locally irregular chromatic in...

In this paper, we propose a new approach for the computation of the statistics of knowledge graphs. We introduce a schema graph that represents the main framework for the computation of the statistics. The core of the procedure is an algorithm that determines the sub-graph of the schema graph affected by the insertion of one triple into the triple-...

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum value of $k$ for which such a decomposition of $G$ exists is the odd chromatic index, $\chi_{o}'(G)$, introduced b...

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)∪E(G) is called the mixed metric dimension of G, and it is denoted by mdim(G). In [12] it was conjectured that every graph G with cyclomatic number c(G) satisfies mdim(G)≤L1(G)+2c(G) where L1(G) is the number of leaves in G. It is already p...

In a connected graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)∪E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicyclic graph G which is derived from the structure of G. We further consider graphs G with edg...

The vertex (resp. edge) metric dimension of a connected graph G; denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S in V (G) which distinguishes all pairs of vertices (resp. edges) in G: Bounds dim(G) <= L(G)+2c(G) and edim(G) <= L(G) + 2c(G); where c(G) is the cyclomatic number in G and L(G) depends on the number of leav...

Let [k]={1,2,…,k}, where the elements of [k] are called colors. Supposing f is a function which assigns a subset of [k] to each vertex of a graph G, if each vertex which is assigned the empty set has all k colors in its neighborhood, then f is called a k-rainbow dominating function for G. If f satisfies the additional condition that for every v∈V(G...

In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G: In [19] we determined the vertex (resp. edge) metric d...

Given a connected graph G, the metric (resp. edge metric) dimension of G is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of G by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifically, we...

A set S of vertices in a graph G is a dominating set if every vertex of G is in S or is adjacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The domination number γ ( G ) of G is the minimum cardinality of a dominating set in G, while the independent domination number i ( G ) of G is the mini...

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V(G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the ver...

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that are d...

For a given graph G, the metric and edge metric dimensions of G, dim(G) and edim(G), are the cardinalities of the smallest possible subsets of vertices in V(G) such that they uniquely identify the vertices and the edges of G, respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparabl...

For a given graph $G$, the metric and edge metric dimensions of $G$, $\dim(G)$ and ${\rm edim}(G)$, are the cardinalities of the smallest possible subsets of vertices in $V(G)$ such that they uniquely identify the vertices and the edges of $G$, respectively, by means of distances. It is already known that metric and edge metric dimensions are not i...

The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First...

Set containment operations form an important tool in various fields such as information retrieval, AI systems, object-relational databases, and Internet applications. In the paper, a set-trie data structure for storing sets is considered, along with the efficient algorithms for the corresponding set containment operations. We present the mathematic...

A nut graph is a simple graph whose adjacency matrix is singular with $1$-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each $d \in \{3,4,\ldots,11\}$ all values $n$ such that there exists a $d$-regular nut graph of order $n$. In the present paper, we determine all values $n...

The arithmetic–geometric index (AG(G)) was recently introduced as a modification of the well-known geometric–arithmetic index (GA(G)). This paper reports results on searching for extremal AG-graphs for various classes of simple graphs. Additionally, relations between these two indices are elaborated. Results on combinations AG+GA, AG−GA, AG · GA, a...

The principal aim of this article is to initiate a study of the following coloring notion for digraphs. An odd k-edge coloring of a general digraph (directed pseudograph) D is a (not necessarily proper) coloring of its edges with at most k colors such that for every vertex v and color c holds: if c is used on the set ∂D(v) of edges incident with v,...

A homomorphism of a signed graph $(G, \sigma)$ to $(H, \pi)$ is a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ such that adjacencies, incidences and the product of signs of closed walks are preserved. Motivated by reformulations of the $k$-coloring problem in this language, and specially in connection with result...

A graph is even (resp. odd) if all its vertex degrees are even (resp. odd). We consider edge coverings by prescribed number of even and/or odd subgraphs. In view of the 8-Flow Theorem, a graph admits a covering by three even subgraphs if and only if it is bridgeless. Coverability by three odd subgraphs has been characterized recently [Petruševski,...

Nanotubical graphs are obtained by wrapping a hexagonal grid, and then possibly closing the tube with caps. In this paper we derive asymptotics for generalized Wiener index for such graphs. More generally, we show that if $I^{\lambda}(G)=\sum_{u\ne v}f(u,v)\dist^{\lambda}(u,v)$, where $\lambda\ge -1$ and $f(u,v)$ is a nonnegative symmetric function...

Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in G, and the eccentric complexity, Cec(G), is the number of different eccentricities in G. In this paper, we prove that for every integer c there are infinitely many graphs G such that CW(G)−Cec(G)=c. Mo...

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)U E(G) is called the mixed metric dimension of G, and it is denoted by mdim(G). In [12] it was conjectured that for a graph G with cyclomatic number c(G) it holds that mdim(G) <= L1(G) + 2c(G) where L1(G) is the number of leaves in G. It is...

There are three different approaches for constructing nanotori in the literature: one with three parameters suggested by Altshuler, another with four parameters used mostly in chemistry and physics after the discovery of fullerene molecules, and one with three parameters used in interconnecting networks of computer science known under the name gene...

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $G$ is in $S$ or is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set in $G$, while the independent domination number $i...

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G)[E(G) is called the mixed metric dimension of G. In this paper we ?rst establish the exact value of the mixed metric dimension of a unicycic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint...

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the ve...

The importance of individuals and groups in networks is modeled by various centrality measures. Additionally, Freeman’s centralization is a way to normalize any given centrality or group centrality measure, which enables us to compare individuals or groups from different networks. In this paper, we focus on degree-based measures of group centrality...

Wiener index, which is defined as the sum of distances between all unordered pairs of vertices, is one of the oldest and most popular molecular descriptors. It is known that among graphs on n vertices that have just one block, the n-cycle has the biggest Wiener index. In a previous work of the same authors, it was shown that among all graphs on n v...

A (finite) graph is odd if all its vertices have odd degrees. The principal aim of this survey is to present the current state of research on covers and decompositions of graphs into fewest possible number of odd subgraphs. Given a graph G, the parameters χo′(G) and covo(G) denote, respectively, the minimum size of a decomposition and cover of G co...

Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifical...

A graph G is called an ℓ-apex tree if there exist a vertex subset A ⊂ V ( G ) with cardinality ℓ such that G − A is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalized Wiener in...

The weighted Szeged index ( w S z ) has gained considerable attention recently because of its unusual mathematical properties. Searching for a tree (or trees) that minimizes the w S z is still going on. Several structural details of a minimal tree were described. Here, it is shown a surprising property that these trees have maximum degree at most 1...

Wiener index, defined as the sum of distances between all unordered pairs of vertices, is one of the most popular molecular descriptors. It is well known that among 2-vertex connected graphs on n≥3 vertices, the cycle Cn attains the maximum value of Wiener index. We show that the second maximum graph is obtained from Cn by introducing a new edge th...

Recently the notion of $k$-rainbow total domination was introduced for a graph $G$, motivated by a desire to reduce the problem of computing the total domination number of the generalized prism $G \Box K_k$ to an integer labeling problem on $G$. In this paper we further demonstrate usefulness of the labeling approach, presenting bounds on the rainb...

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with 5 colours such that for every edge e, the set of colours assigned to the edges adjacent to e has cardinality either 2 or 4, but not 3. We prove that every bridgeless cubic graph $G$ admits an edge-colouring with 4 colours such that at most $8/15...

The Wiener index (the distance) of a connected graph is the sum of distances between all pairs of vertices. In this paper, we study the maximum possible value of this invariant among graphs on n vertices with fixed number of blocks p. It is known that among graphs on n vertices that have just one block, the n-cycle has the largest Wiener index. And...

A well-known inequality between the Szeged and Wiener indices says that Sz(G)=∑e=ij∈E(G)ne(i)ne(j)≥∑{u,v}d(u,v)=W(G) for every graph G. In the past, variable variations of the standard topological indices were defined. Following this line, we study a natural generalisation of the above inequality, namely ∑e=ij∈E(G)(ne(i)ne(j))α≥∑{u,v}d(u,v)α. We sh...

Let G be a graph. The Graovac–Pisanski index is defined as [Formula Presented], where Aut(G) is the group of automorphisms of G. This index is considered to be an extension of the original Wiener index, since it takes into account not only the distances, but also the symmetries of the graph. In this paper, for each n we find all trees on n vertices...

Consider the smallest distance among k distinct vertices in a graph. How large can it be? Let G be a connected graph on n vertices, let k satisfy 3 ≤ k < n and let v 1 ,v 2 …,v k be distinct vertices in G. We prove that there are i and j, 1 ≤ i < j ≤ k, such that d G (v i ,v j )≤[Formula presented]. Moreover, for every k and n the subdivided k-star...

The Wiener index of a graph is one of the most recognised and very well-researched topological indices, i.e. graph theoretic invariants of molecular graphs. Nonetheless, some interesting questions remain largely unsolved despite being easy to state and comprehend. In this paper, we investigate a long-standing question raised by Plesník in 1984, nam...

A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring usi...

The atom-bond connectivity (ABC) index is one of the most actively studied degree-based graph invariants, which are found in a vast variety of chemical applications. This work is devoted to establishing some extremal results regarding a variant of the ABC index, the so-called multiplicative atom-bond connectivity index (ABCΠ), which for a graph G i...

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with $5$ colours such that for every edge $e$, the set of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or $4$, but not $3$. We prove that every bridgeless cubic graph $G$ admits an edge-colouring with $4$ colours such that...

The Wiener index (the distance) of a connected graph is the sum of distances between all pairs of vertices. In this paper, we study the maximum possible value of this invariant among graphs on $n$ vertices with fixed number of blocks $p$. It is known that among graphs on $n$ vertices that have just one block, the $n$-cycle has the largest Wiener in...

Wiener index, defined as the sum of distances between all unordered pairs of vertices, is one of the most popular molecular descriptors. It is well known that among 2-vertex connected graphs on $n\ge 3$ vertices, the cycle $C_n$ attains the maximum value of Wiener index. We show that the second maximum graph is obtained from $C_n$ by introducing a...

An odd graph is a graph whose vertex degrees are all odd. As introduced by Pyber in 1991, an odd edge‐covering of graph is a family of odd subgraphs that cover its edges. The minimum size of such family is denoted by . Answering a question raised by Pyber, Mátrai proved in 2006 that for every simple graph . In this study, we characterize the same i...

An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index ($\textrm{sz}(G)$). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular, we proved that the star is a tree having the maximal $\textrm{sz}(G)$. Finding a tree with the minimal $\textr...

Wiener index, defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the oldest and the most popular molecular descriptors. In the paper, we would like to point to an “overlooked” problem of determining the minimum value of this index and corresponding extremes among chemical graphs with prescribed number of v...

The Wiener index W(G) of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991, Šoltés posed the problem of finding all graphs G such that the equality W(G)=W(G−v) holds for all their vertices v. Up to now, the only known graph with this property is the cycle C11. Our main object of study is a relaxed ve...

In the 1-in port model, every vertex of a synchronous network can receive at most one message in each time unit. We consider simultaneous broadcasting of multiple messages from the same source or from distinct sources in such networks with an additional restriction that every received message can be sent out to neighbors only in the next time unit...

Nanotubical graphs are obtained by wrapping a hexagonal grid, and then possibly closing the tube with caps. In this paper we show that the asymptotics for Balaban, Sum-Balaban, and Harary indices for all nanotubical graphs of type $(k,l)$ on $n$ vertices are $\frac{9\pi(k+l)}{2n}$, $\frac{9\sqrt 2}2\sqrt{k+l}\cdot\log(1+\sqrt 2)$ and $(k+l)n\log (n...

We propose and investigate a new bond-additive structural invariant as a measure of peripherality in graphs. We first determine its extremal values and characterize extremal trees and unicyclic graphs. Then we show how it can be efficiently computed for large classes of chemically interesting graphs using a variant of the cut method introduced by K...