arXiv subjects

Martin Knor

Publications and source records attributed to Martin Knor.

At least 19 recordsLinked to original sources

arXiv2026

On the odd independence number of the Queen graph

A set S of vertices of a graph is odd independent if it is independent and every vertex outside S has either zero or an odd number of neighbors in S. The largest size of such a set is the odd independence number alpha_od. Caro, Petrusevski, Skrekovski and Tuza [2] conjectured that alpha_od = 1 for every finite Queen graph. They also asked whether the infinite Queen graph has alpha_od = 1 or alpha_od = infinity. We prove that alpha_od = 1 in both cases. In particular, in the case of an infinite board we prove that alpha_od = 1 holds on the quarter plane and on the whole plane.

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arXiv2026

Counterexamples to two conjectures on (1, 2)-domination in cubic graphs

Let G be a cubic graph of order n. The induced cycles vertex number cind(G) is the largest size of a vertex set that induces a 2-regular subgraph of G. By gamma_1,2(G) we denote the (1,2)-domination number of G. Erves and Tepeh introduced the trilobite graphs T_n, which satisfy gamma_1,2(T_n) > cind(T_n). They stated two conjectures, the first of which says that every cubic graph G with cind(G) >= n/2 + 2 satisfies gamma_1,2(G) <= cind(G). The second says that a connected cubic graph G satisfies gamma_1,2(G) > cind(G) if and only if G is a trilobite. We show that both conjectures are false. A computer search finds counterexamples that are not trilobites already for n = 18, 20 and 22. We also construct an infinite family H(k) of order n = 20 + 4k. For every k >= 0 we prove that cind(H(k)) = n/2 + 2 and gamma_1,2(H(k)) = n/2 + 3. Hence both conjectures fail for infinitely many orders n.

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arXiv2026

A complete characterization of maximally {\sigma}-irregular trees with prescribed maximum degree

The sigma-irregularity of a graph G = (V, E) is defined as the sum, over all edges uv in E, of (d(u) - d(v))^2, where d(u) denotes the degree of vertex u. A tree on n vertices with maximum degree Delta is called maximal if it attains the greatest possible sigma-irregularity among all such trees. The maximal trees are already known for chemical trees (Delta <= 4) and for Delta = 5. In this paper, we characterize the maximal trees for every Delta >= 6 when n >= Delta(Delta - 1) + 1. We introduce three families of trees, T'{n,Delta}, T''{n,Delta}, and T'''{n,Delta}. All trees within the same family have the same sigma-irregularity, and we derive an explicit formula for the value attained by each family. Comparing these formulas determines which family has the greatest sigma-irregularity for given n and Delta. We then prove that a tree is maximal if and only if it belongs to a family attaining this greatest value. In contrast to the case Delta <= 5, the two natural candidate families T'{n,Delta} and T''{n,Delta} are not sufficient, since for every Delta >= 7 and every n congruent to 3 modulo Delta, the trees in T'''{n,Delta} have strictly greater sigma-irregularity.

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arXiv2026

Counting geodesic paths in graphs

A geodesic is a shortest path which connects a pair of vertices of a graph G. In this paper we define the geodesic subpath number gpn(G) of a graph G as the number of geodesics in G. The number of subtrees and subpaths are already studied in literature, but they are both large quantities. Hence, the geodesic subpath number which is related to these quantities but smaller than both, seems worthy of investigation. We first consider extremal graphs with respect to the geodesic subpath number among all connected graphs on n vertices. This number is minimized by the so called geodetic graphs, i.e. graphs in which each pair of vertices is connected by precisely one geodesic. As for the graphs which maximize the geodesic subpath number, we provide an upper bound on gpn(G) in terms of n and we further consider several graph families which might have a large gpn(G). Yet, their value of gpn(G) still does not attain the established bound, so narrowing the gap remains as an open problem. We also consider the class of cactus graphs on n vertices and k cycles and among them characterize extremal graphs with respect to this new invariant.

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arXiv2025

The subpath number of cactus graphs

The subpath number of a graph G is defined as the total number of subpaths in G, and it is closely related to the number of subtrees, a well-studied topic in graph theory. This paper is a continuation of our previous paper [5], where we investigated the subpath number and identified extremal graphs within the classes of trees, unicyclic graphs, bipartite graphs, and cycle chains. Here, we focus on the subpath number of cactus graphs and characterize all maximal and minimal cacti with n vertices and k cycles. We prove that maximal cacti are cycle chains in which all interior cycles are triangles, while the two end-cycles differ in length by at most one. In contrast, minimal cacti consist of k triangles, all sharing a common vertex, with the remaining vertices forming a tree attached to this joint vertex. By comparing extremal cacti with respect to the subpath number to those that are extremal for the subtree number and the Wiener index, we demonstrate that the subpath number does not correlate with either of these quantities, as their corresponding extremal graphs differ.

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arXiv2025

Invitation to the subpath number

In this paper we count all the subpaths of a given graph G; including the subpaths of length zero, and we call this quantity the subpath number of G. The subpath number is related to the extensively studied number of subtrees, as it can be considered as counting subtrees with the additional requirement of maximum degree being two. We first give the explicit formula for the subpath number of trees and unicyclic graphs. We show that among connected graphs on the same number of vertices, the minimum of the subpath number is attained for any tree and the maximum for the complete graph. Further, we show that the complete bipartite graph with partite sets of almost equal size maximizes the subpath number among all bipartite graphs. The explicit formula for cycle chains, i.e. graphs in which two consecutive cycles share a single edge, is also given. This family of graphs includes the unbranched catacondensed benzenoids which implies a possible application of the result in chemistry. The paper is concluded with several directions for possible further research where several conjectures are provided.

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arXiv2024

Some results on $\sigma_{t}$-irregularity

The $\sigma_{t}$-irregularity (or sigma total index) is a graph invariant which is defined as $\sigma_{t}(G)=\sum_{\{u,v\}\subseteq V(G)}(d(u)-d(v))^{2},$ where $d(z)$ denotes the degree of $z$. This irregularity measure was proposed by R\' {e}ti [Appl. Math. Comput. 344-345 (2019) 107-115], and recently rediscovered by Dimitrov and Stevanovi\'c [Appl. Math. Comput. 441 (2023) 127709]. In this paper we remark that $\sigma_{t}(G)=n^{2}\cdot Var(G)$, where $Var(G)$ is the degree variance of the graph. Based on this observation, we characterize irregular graphs with maximum $\sigma_{t}$-irregularity. We show that among all connected graphs on $n$ vertices, the split graphs $S_{\lceil\frac{n}{4}\rceil, \lfloor\frac{3n}{4}\rfloor }$ and $S_{\lfloor\frac{n}{4}\rfloor, \lceil\frac{3n}{4}\rceil }$ have the maximum $\sigma_{t}$-irregularity, and among all complete bipartite graphs on $n$ vertices, either the complete bipartite graph $K_{\lfloor\frac{n}{4}(2-\sqrt{2})\rfloor, \lceil\frac{n}{4}(2+\sqrt{2})\rceil }$ or $K_{\lceil\frac{n}{4}(2-\sqrt{2})\rceil, \lfloor\frac{n}{4}(2+\sqrt{2})\rfloor }$ has the maximum sigma total index. Moreover, various upper and lower bounds for $\sigma_{t}$-irregularity are provided; in this direction we give a relation between the graph energy $\mathcal{E}(G)$ and sigma total index $\sigma_{t}(G)$ and give another proof of two results by Dimitrov and Stevanovi\'c. Applying Fiedler's characterization of the largest and the second smallest Laplacian eigenvalue of the graph, we also establish new relationships between $\sigma_{t}$ and $\sigma$. We conclude the paper with two conjectures.

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arXiv2024

Extremizing antiregular graphs by modifying total $\sigma$-irregularity

The total $\sigma$-irregularity is given by $ \sigma_t(G) = \sum_{\{u,v\} \subseteq V(G)} \left(d_G(u) - d_G(v)\right)^2, $ where $d_G(z)$ indicates the degree of a vertex $z$ within the graph $G$. It is known that the graphs maximizing $\sigma_{t}$-irregularity are split graphs with only a few distinct degrees. Since one might typically expect that graphs with as many distinct degrees as possible achieve maximum irregularity measures, we modify this invariant to $ \IR(G)= \sum_{\{u,v\} \subseteq V(G)} |d_G(u)-d_G(v)|^{f(n)}, $ where $n=|V(G)|$ and $f(n)>0$. We study under what conditions the above modification obtains its maximum for antiregular graphs. We consider general graphs, trees, and chemical graphs, and accompany our results with a few problems and conjectures.

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arXiv2024

Fault tolerance of metric basis can be expensive

A set of vertices S is a resolving set of a graph G; if for every pair of vertices x and y in G, there exists a vertex s in S such that x and y differ in distance to s. A smallest resolving set of G is called a metric basis. The metric dimension dim(G) is the cardinality of a metric basis of G. The notion of a metric basis is applied to the problem of placing sensors in a network, where the problem of sensor faults can arise. The fault-tolerant metric dimension ftdim(G) is the cardinality of a smallest resolving set S such that S\{s} remains a resolving set of G for every s in S. A natural question is how much more sensors need to be used to achieve a fault-tolerant metric basis. It is known in literature that there exists an upper bound on ftdim(G) which is exponential in terms of dim(G); i.e. ftdim(G) <= dim(G)(1+2^(5dim(G)-1)). In this paper, we construct graphs G with ftdim(G) = dim(G)+2^(dim(G)-1) for any value of dim(G), so the exponential upper bound is necessary. We also extend these results to the k-metric dimension which is a generalization of the fault-tolerant metric dimension. First, we establish a similar exponential upper bound on dim(k+1)(G) in terms of dim(k)(G); and then we show that there exists a graph for which dim(k+1)(G) is indeed exponential. For a possible further work, we leave the gap between the bounds to be reduced.

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arXiv2023

On regular graphs with \v{S}olt\'es vertices

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 vertices. The only such graph known to this day is $C_{11}$. As the original problem appears to be too challenging, several relaxations were studied: one may look for graphs with at least $k$ \v{S}olt\'es vertices; or one may look for $\alpha$-\v{S}olt\'es graphs, i.e. graphs where the ratio between the number of \v{S}olt\'es vertices and the order of the graph is at least $\alpha$. Note that the original problem is, in fact, to find all $1$-\v{S}olt\'es graphs. We intuitively believe that every $1$-\v{S}olt\'es graph has to be regular and has to possess a high degree of symmetry. Therefore, we are interested in regular graphs that contain one or more \v{S}olt\'es vertices. In this paper, we present several partial results. For every $r\ge 1$ we describe a construction of an infinite family of cubic $2$-connected graphs with at least $2^r$ \v{S}olt\'es vertices. Moreover, we report that a computer search on publicly available collections of vertex-transitive graphs did not reveal any $1$-\v{S}olt\'es graph. We are only able to provide examples of large $\frac{1}{3}$-\v{S}olt\'es graphs that are obtained by truncating certain cubic vertex-transitive graphs. This leads us to believe that no $1$-\v{S}olt\'es graph other than $C_{11}$ exists.

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arXiv2023

Selected topics on Wiener index

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. Contemp., 2016) in which several interesting open questions on the topic were outlined. Here we collect answers gathered so far, give further insights on the topic of extremal values of Wiener index in different settings, and present further intriguing problems and conjectures.

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arXiv2022

On chromatic vertex stability of 3-chromatic graphs with maximum degree 4

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

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arXiv2022

Remarks on the vertex and the edge metric dimension of 2-connected graphs

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 without leaves distinct from cycles, and moreover all leafless cacti which attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper we focus on Theta graphs, as the most simple 2-connected graphs distinct from cycle, and show that the the upper bound 2c(G) - 1 holds for both metric dimensions of Theta graphs and we characterize all Theta graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2c(G) - 1 in the class of leafless graphs besides already known extremal cacti and extremal Theta graphs mentioned here.

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arXiv2022

On maximum Wiener index of directed grids

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 Šumenjak et al. proved that the maximum Wiener index of a digraph, which is obtained by orienting the edges of $L_n$, is obtained when all layers isomorphic to one factor are directed paths directed in the same way except one (corresponding to an endvertex of the other factor) which is a directed path directed in the opposite way. Then they conjectured that the natural generalization of this orientation to $G_{m,n}$ will attain the maximum Wiener index among all orientations of $G_{m,n}$. In this paper we disprove the conjecture by showing that a comb-like orientation of $G_{m,n}$ has significiantly bigger Wiener index.

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arXiv2021

The Graovac-Pisanski index of connected bipartite graphs with applications to hydrocarbon molecules

The Graovac-Pisanski index, also called the modified Wiener index, was introduced in 1991 and represents an extension of the original Wiener index, because it considers beside the distances in a graph also its symmetries. Similarly as Wiener in 1947 showed the correlation of the Wiener indices of the alkane series with the boiling points, in 2018 the connection between the Graovac-Pisanski index and the melting points of some hydrocarbon molecules was established. In this paper, we prove that the Graovac-Pisanski index of any connected bipartite graph as well as of any connected graph on an even number of vertices is an integer number. These results are applied to some important families of hydrocarbon molecules. By using a computer programme, the graphs with a non-integer Graovac-Pisanski index on at most nine vertices are counted. Finally, an infinite class of unicyclic graphs with a non-integer Graovac-Pisanski index is described.

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arXiv2021

A note on the metric and edge metric dimensions of 2-connected graphs

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 in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph $G$ such that $\dim(G)=a$ and ${\rm edim}(G)=b$ for every pair of integers $a,b$, where $4\le b<a$. For this we use subdivisions of complete graphs, whose metric dimension is in some cases smaller than the edge metric dimension. Along the way, we present an upper bound for the metric and edge metric dimensions of subdivision graphs under some special conditions.

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arXiv2021

On $12$-regular nut graphs

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 $n$ for which a $12$-regular nut graph of order $n$ exists. We also present a result by which there are infinitely many circulant nut graphs of degree $d \equiv 0 \pmod 4$ and no circulant nut graph of degree $d \equiv 2 \pmod 4$.

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arXiv2020

Domination versus independent domination in regular graphs

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 minimum cardinality of an independent dominating set in $G$. We prove that for all integers $k \geq 3$ it holds that if $G$ is a connected $k$-regular graph, then $\frac{i(G)}{γ(G)} \leq \frac{k}{2}$, with equality if and only if $G = K_{k,k}$. The result was previously known only for $k\leq 6$. This affirmatively answers a recent question of Babikir and Henning.

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