# Tomislav DoslicUniversity of Zagreb · Department of Mathematics (GRAD)

Tomislav Doslic

Professor

## About

131

Publications

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Citations since 2016

## Publications

Publications (131)

The imbalance of an edge in a graph is defined as the absolute value of the difference of the degrees of its end-vertices. The irregularity of a simple graph G is defined as the sum of the imbalances of all edges of G. It is used as a measure to quantify the deviation of a graph from being regular. In this paper, a sharp lower bound on the irregula...

In this paper we explore two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana's cow sequence and provide combinatorial proofs for several known identities about those numbers.

A graph G with a perfect matching is k-extendable if it has a matching of size k and if every such matching can be extended to a perfect matching in G. It is well known that (3,6)-fullerene graphs are not 2-extendable. In this contribution we examine what can be salvaged and under what conditions can a matching of size two be extended to a perfect...

The Mostar index is a recently introduced bond-additive distance-based graph invariant that measures the degree of peripherality of particular edges and of the graph as a whole. It attracted considerable attention, both in the context of complex networks and in more classical applications of chemical graph theory, where it turned out to be useful a...

The Sombor index is a recently introduced graph-theoretical invariant of the bond-additive type. It is known that it takes integer values for bipartite semi-regular graphs whose degrees appear as two smaller elements in a Pythagorean triple. In this note we show that it can have integer values also for graphs with more complicated structure and con...

This paper is concerned with a recently introduced graph invariant, namely the Sombor index. Some bounds on the Sombor index are derived and then utilized to establish additional bounds by making use of the existing results. One of the direct consequences of one of the obtained bounds is that the cycle graph Cn attains the minimum Sombor index amon...

We present explicit formulas for the Laplacian Szeged eigenvalues of paths, grids, C 4-nanotubes and of Cartesian products of paths with some other simple graphs. A number of open problems is listed.

A matching M of a graph G is maximal if it is not a proper subset of any other matching in G. Maximal matchings are much less known and researched than their maximum and perfect counterparts. In this paper we present the recurrences and generating functions for the sequences enumerating maximal matchings in two classes of chemically interesting lin...

A subgraph H of a graph G with perfect matching is nice if G−V(H) has perfect matching. It is well-known that all fullerene graphs have perfect matchings and that all fullerene graphs contain some small connected graphs as nice subgraphs. In this contribution, we consider fullerene graphs arising from smaller fullerenes via the leapfrog transformat...

Let G be a graph with a perfect matching. A subgraph H of G is nice if G-V(H) still has a perfect matching. In a chemical context, nice subgraphs of molecular graphs serve as mathematical models of addition patterns in the corresponding molecules such that the rest of the molecule still has a resonant structure. In this contribution we start from t...

A perfect star packing in a graph G is a spanning subgraph of G whose every component is isomorphic to the star graph K 1,3. We investigate which fullerene graphs allow such packings. We also consider generalized fullerene graphs and packings of other graphs into classical and generalized fullerenes. Several open problems are listed.

For a connected graph G, the Mostar index Mo(G) and the irregularity irr(G) are defined as Mo(G) = ∑ uv∈E(G) |n u − n v | and irr(G) = ∑ uv∈E(G) |d u − d v |, respectively, where d u is the degree of the vertex u of G and n u denotes the number of vertices of G which are closer to u than to v for an edge uv. In this paper we focus on the difference...

We introduce and investigate a new topological index, the Lanzhou index, and show that it outperforms several existing indices on some benchmark datasets recommended by the International Academy of Mathematical Chemistry. We determine its extremal values and characterize extremal graphs, trees, and restricted trees. Nothing speaks more on the quali...

The paper is concerned with the weighted Harary indices, namely the multiplicatively weighted Harary index HM and the additively weighted Harary index HA. For a simple connected graph G with n vertices, m edges and k cut edges, sharp upper bounds on HM(G) and HA(G) are derived and the corresponding extremal graphs are characterized. From one of the...

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...

In this paper we generalize and unify results of several recent papers by presenting explicit formulas for the number of spanning trees in a class of unbranched polycyclic polymers. From these formulas we immediately deduce the asymptotic behavior of the number of spanning trees, and, as a consequence, we obtain combinatorial proofs of some identit...

A matching M in a graph G is maximal if no other matching of G has M as a proper subset. The saturation number of G is the cardinality of any smallest maximal matching in G. In this paper we investigate saturation number for several classes of square and hexagonal lattice animals.

We apply the concepts of importance and redundancy to compute and analyze the partition of \(\pi \)-electrons among faces of actual and potential polyhedral carbon clusters. In particular, we present explicit formulas and investigate asymptotic behavior of total and average \(\pi \)-electron content of all faces of prisms and n-barrels. We also dis...

Let M(G) denote the set of all maximal matchings in a simple graph G, and f : M(G) → {0, 1} |E(G)| be the characteristic function of maximal matchings of G. Any set S ⊆ E(G) such that f | S is an injection is called a global forcing set for maximal matchings in G, and the cardinality of smallest such S is called the global forcing number for maxima...

We apply the concepts of importance and redundancy to compute and analyze the partition of π-electrons among faces of nanotubical fullerenes. We also discuss the deviation from uniform distribution as a potential predictor of fullerene stability. © 2018 University of Kragujevac, Faculty of Science. All rights reserved.

Let G be a simple connected graph. The eccentric complexity of graph G is introduced as the number of different eccentricities of its vertices. A graph with eccentric complexity equal to one is called self-centered. In this paper, we study the eccentric complexity of graph under several graph operations such as complement of graph, line graph, Cart...

We show how generalized Zagreb indices M k 1 (G) can be computed by using a simple graph polynomial and Stirling numbers of the second kind. In that way we explain and clarify the meaning of a triangle of numbers used to establish the same result in an earlier reference.

A fullerene G is a 3-regular 3-connected plane graph consisting of only pentagonal and hexagonal faces. The resonance graph R(G) of G reflects the structure of the set of its perfect matchings. In this paper we show that if a connected component of the resonance graph of a fullerene is not a path, then this component without vertices of degree one...

The inverse sum indeg index is a recently-introduced bond-additive
descriptor that was selected by Vukicevic and Gasperov
(2010) as a significant predictor of total surface area of octane
isomers. In this paper, we present several sharp upper and lower
bounds on the inverse sum indeg index in terms of some graph
parameters such as the order, size,...

We present explicit formulas for augmented eccentric connectivity indices of several classes of grid graphs that arise via Cartesian product. We also explore their asymptotic behavior and compute the compression ratios for considered graphs.

The Hosoya index () Z G of a graph G is the total number of matchings in G. We present explicit formulas for the Hosoya indices of several classes of graphs that arise from simpler graphs by repeating application of two simple operations.

Singly and doubly vertex-weighted Wiener polynomials are generalizations of both vertex-weighted Wiener numbers and the ordinary Wiener polynomial. In this paper, we show how the vertex-weighted Wiener polynomials of a graph change with subdivision operators, and apply our results to obtain vertex-weighted Wiener numbers.

A matchingM in a graph G is maximal if it cannot be extended to a larger matching in G. In this paper we show how several chemical and technical problems can be successfully modeled in terms of maximal matchings. We introduce the maximal matching polynomial and study its basic properties. Then we enumerate maximal matchings in several classes of gr...

A connected planar graph is called m-generalized fullerene if two of its faces are m-gons and all other faces are pentagons and hexagons. In this paper we first determine some structural properties of m-generalized fullerenes and then use them to obtain new results on the enumerative aspects of perfect matchings in such graphs. We provide both uppe...

In this paper we present explicit formulas for domination numbers of equidistant $m$-cactus chains and find the corresponding minimum dominating sets. For an arbitrary $m$-cactus chain, we establish the lower and the upper bound for its domination number. We find some extremal chains with respect to this graph invariant.

We use recently obtained lower bounds on the independence number of fullerene graphs to settle in affirmative a conjecture of the Graffiti software about the relationship between the independence number and the face independence number of a fullerene graph. We also consider another Graffiti conjecture, concerned with the relationship of the indepen...

We show that a factor-critical graph of order has exactly near-perfect matchings if and only if it is a connected graph whose blocks are all odd cycles. This characterizes the factor-critical graphs with the minimum number of near-perfect matchings.

A matching $M$ of a graph $G$ is maximal if it is not a proper subset of any
other matching in $G$. Maximal matchings are much less known and researched
than their maximum and perfect counterparts. In particular, almost nothing is
known about their enumerative properties. In this paper we present the
recurrences and generating functions for the seq...

In this paper we consider the number of dominating sets in cactus chains with triangular and square blocks. We derive and solve the recurrences satisfied by those quantities and investigate their asymptotic behavior. In triangular case we also refine the counting by computing the bivariate generating function. As a corollary, we compute the expecte...

Fullerene graphs are 3-connected cubic planar graphs with only pentagonal and hexagonal faces. Nanotubes are special type of fullerene graphs determined by a vector (p; q). We show that the diameter of a (p; q)-nanotubical fullerene graph is essentially n=(p+q). In addition, we determine the diameter of (9; 0)-isolated pentagon nanotubes.

The saturation number of a graph is the cardinality of any smallest max-imal matching in the graph. We present explicit results and tight asymptotic bounds on the saturation number of several classes of benzenoid graphs.

The signless Laplacian Estrada index of a graph $G$ is defined as
$SLEE(G)=\sum^{n}_{i=1}e^{q_i}$ where $q_1, q_2, \ldots, q_n$ are the
eigenvalues of the signless Laplacian matrix of $G$. In this paper, we show
that there are exactly two tricyclic graphs with the maximal signless Laplacian
Estrada index.

The Kepler-Bouwkamp constant is defined as the limit of radii of a sequence of con-centric circles that are simultaneously inscribed in a regular n-gon and circumscribed around a regular (n + 1)-gon for n ≥ 3. The outermost circle, circumscribed around an equilateral triangle, has radius 1. We investigate what happens when the number of sides of re...

Let G be a simple connected graph. The distance between the edges g and f ∈ E(G) is defined as the distance between the corresponding vertices g and f in the line graph of G. The edge-Wiener index of G is defined as the sum of such distances between all pairs of edges of the graph. Let G1+G2 and G1 G2 be the join and the corona of graphs G1 and G2,...

We present explicit formulas for the values of eccentric connec-tivity index for several families of composite graphs. The results are applied to some graphs of chemical interest, such as C4 nanotubes and nanotori.

We consider an infinite lattice and show that its total eccentricity can be used to deduce its dimensionality. The new definition is consistent with both the classical approach and the Wiener dimensionality, defined recently in a paper by Ante Graovac and his coauthors [14].

The Narumi-Katayama index of a graph G, denoted by N K(G), is equal to the product of degrees of vertices of G. In this paper we investigate its behavior under several binary operations on graphs. We present explicit formulas for its values for composite graphs in terms of its values for operands and some auxiliary invariants. We demonstrate applic...

A matching M in a graph G is a set of edges of G such that no two edges from M have a vertex in common [1]. A matching M is maximal if it cannot be extended to a larger matching. Saturation number of a graph G is the minimum possible size of a maximal matching in G.In this work we are concerned with determining the saturation number for several cla...

We consider several classes of planar polycyclic graphs and derive recurrences satisfied by their W.T. Tutte polynomials [Can.J. Math. 6, 80–91 (1954; Zbl 0055.17101); Proc. 5th Br. Comb. Conf., Aberdeen 1975, 605–635 (1976; Zbl 0339.05105)]. The recurrences are then solved by computing the corresponding generating functions. As a consequence we ob...

We compute several eccentricity-related invariants for small single-defect nanocones and then fit a cubic polynomial through the computed values in order to obtain explicit formulas.

In this paper we present explicit formulas for the eccentric connectivity index of three classes of chain hexagonal cacti. Further, it is shown that the extremal chain hexagonal cacti with respect to the eccentric connectivity index belong to one of the considered types. Some open problems and possible directions of further research are mentioned i...

Some new relations have been established between Wiener indices, stability numbers and clique numbers for several classes of composite graphs that arise via graph products. For three of considered operations we show that they make a multiplicative pair with the clique number.

We introduce a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. After establishing basic mathematical properties of the new invariant, we proceed by finding the extremal graphs and investigating its behavior under several standard graph products.

The icosahedral isomer of C 6 0 has the maximum smallest eigenvalue amongst all fullerenes on 60 or more vertices. This settles in affirmative a decade old conjecture.

In the present study we investigate some general problems concerning the degeneracy of widely used topological indices (graph invariants), and we propose a novel family of molecular descriptors characterized by a decreased degeneracy level. A special feature of topological indices of novel type is that they take into account the degrees of vertices...

Fullerene graphs are $3$-connected $3$-regular planar graphs with only pentagonal and hexagonal faces. We show that the diameter of a fullerene graph $G$ of order $n$ is at least $fracsqrt24n - 15 - 36$ and at most $n/5 + 1$. Moreover, if $G$ is not a $(5,0)$-nanotube its diameter is at most $n/6 + 5/2$. As a consequence, we improve the upper bound...

A thread in a graph G is any maximal connected subgraph induced by a set of vertices of degree 2 in G. A string in G is a subgraph induced by a thread and the vertices adjacent to it. A graph G consists of s strings if it can be represented as a union of s strings so that any two strings have at most two vertices in common. We compute several recen...

We use the concept of k-domination, where k≥1. We determine minimum k-dominating sets and k-domination numbers of three special types of hexagonal cactus chains. Those are para-, meta- and ortho-chains. For an arbitrary hexagonal chain G h of length h≥1 we establish the lower and the upper bound for k-domination number γ k . As a consequence, we fi...

Zagreb indices belong to better known and better researched topological indices. We investigate here their ability to discriminate among benzenoid graphs and arrive at some quite unexpected conclusions. Along the way we establish tight (and sometimes sharp) lower andupper bounds on various classes of benzenoids.

We investigate enumerative properties of unbranched polyphenylene chains. In particular, we find exact formulas for the numbers of matchings and independent sets of given cardinalities in three types of uniform chains. Further, we show that two of those three types are extremal with respect to the number of considered structures among all chains of...

We introduce a family of invariants defined in terms of positive functions of degrees of vertices in a graph. A member of the family that measures the average degree of neighbors of vertices in a graph is then investigated for the predictive potential for stability in the class of generalized fullerenes.

We have computed the diameters of all fullerene graphs on $20 \leq n \leq
120$ vertices and of all fullerene graphs with isolated pentagons on $60
\leq n \leq 146$ vertices. The results are used to asses the quality of
recently obtained linear upper bounds and square root-type lower bounds.
It seems that the fullerenes with large diameters are exce...

Recently introduced Zagreb coindices are a generalization of classical Zagreb indices of graphs. In this paper we determine the extremal values of these new topological invariants over some special classes of graphs. The extremal graphs are also presented.

The modified eccentricity connectivity polynomial of a connected graph G is defined as Ξ(G;x)=∑ u∈V(G) d G (u)x ε G ' (u) , where ε G ' (u)=∑ v∈N G (u) ε G (v) and d G (u) is the degree of u in G. In this paper the modified eccentric connectivity polynomial is computed for several classes of composite graphs.

A new class of distance-based molecular structure descriptors is put forward, aimed at eliminating a general shortcoming of the Wiener and Wiener-type indices, namely that the greatest contributions to their numerical values come from vertex pairs at greatest distance. The Q-indices, considered in this work, consist of contributions of vertex pairs...

If G=(VE) is a molecular graph and d u is the degree of vertex u, then the first and second Zagreb indices are ∑ u∈V d u 2 and ∑ uv∈E d u d v ,respectively. These molecular structure descriptors, introduced in the 1970s, have been much studied. Yet, a number of their properties that seem to have evaded attention so far are established in this work...

We use a result from the theory of geometric representation of graphs to show that the separator of a fullerene graph on n vertices cannot exceed 24-n, thus improving the best currently known upper bound of 1-3/n. The result is then combined with a recently established upper bound on the smallest eigenvalue of fullerene graphs to show that there ar...

We present explicit formulae for the eccentric connectivity index of zigzag and armchair hexagonal belts and the corresponding open chains.

We present explicit formulas for the total number of conjugated circuits of a given length in polyacene and fibonacene chains and analyze the asymptotic behavior of the expected number of conjugated circuits in long chains of the considered type.