Bounds on some monotonic topological indices of bipartite graphs with a given number of cut edges

Let $I(G)$ be a topological index of a graph. If $I(G+e)<I(G)$ (or $I(G+e)>I(G)$, respectively) for each edge $e\not\in G$, then $I(G)$ is monotonically decreasing (or increasing, respectively) with the addition of edges. In this article, we present lower or upper bounds for some monotonic topological indices, including the Wiener index, the hyper-Wiener index, the Harary index, the connective eccentricity index, the eccentricity distance sum of bipartite graphs in terms of the number of cut edges, and characterize the corresponding extremal graphs, respectively

Enumeration of bipartite graphs and bipartite blocks

Using the theory of combinatorial species, we compute the cycle index for bipartite graphs, which we use to count unlabeled bipartite graphs and bipartite blocks

Wiener Index, Hyper-wiener Index, Harary Index and Hamiltonicity of graphs

In this paper, we discuss the Hamiltonicity of graphs in terms of Wiener index, hyper-Wiener index and Harary index of their quasi-complement or complement. Firstly, we give some sufficient conditions for an balanced bipartite graph with given the minimum degree to be traceable and Hamiltonian, respectively. Secondly, we present some sufficient conditions for a nearly balanced bipartite graph with given the minimum degree to be traceable. Thirdly, we establish some conditions for a graph with given the minimum degree to be traceable and Hamiltonian, respectively. Finally, we provide some conditions for a $k$-connected graph to be Hamilton-connected and traceable for every vertex, respectively

Signatures, lifts, and eigenvalues of graphs

We study the spectra of cyclic signatures of finite graphs and the corresponding cyclic lifts. Starting from a bipartite Ramanujan graph, we prove the existence of an infinite tower of $3$-cyclic lifts, each of which is again Ramanujan.Comment: 12 pages, 3 figure

Self-similarity of graphs

An old problem raised independently by Jacobson and Sch\"onheim asks to determine the maximum $s$ for which every graph with $m$ edges contains a pair of edge-disjoint isomorphic subgraphs with $s$ edges. In this paper we determine this maximum up to a constant factor. We show that every $m$-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least $c (m\log m)^{2/3}$ edges for some absolute constant $c$, and find graphs where this estimate is off only by a multiplicative constant. Our results improve bounds of Erd\H{o}s, Pach, and Pyber from 1987.Comment: 15 page

More bounds for the Grundy number of graphs

A coloring of a graph $G=(V,E)$ is a partition $\{V_1, V_2, \ldots, V_k\}$ of $V$ into independent sets or color classes. A vertex $v\in V_i$ is a Grundy vertex if it is adjacent to at least one vertex in each color class $V_j$ for every $j<i$. A coloring is a Grundy coloring if every vertex is a Grundy vertex, and the Grundy number $\Gamma(G)$ of a graph $G$ is the maximum number of colors in a Grundy coloring. We provide two new upper bounds on Grundy number of a graph and a stronger version of the well-known Nordhaus-Gaddum theorem. In addition, we give a new characterization for a $\{P_{4}, C_4\}$-free graph by supporting a conjecture of Zaker, which says that $\Gamma(G)\geq \delta(G)+1$ for any $C_4$-free graph $G$.Comment: 12 pages, 1 figure, accepted for publication in Journal of Combinatorial Optimizatio

The 1-2-3 Conjecture and related problems: a survey

The 1-2-3 Conjecture, posed in 2004 by Karonski, Luczak, and Thomason, is as follows: "If G is a graph with no connected component having exactly 2 vertices, then the edges of G may be assigned weights from the set {1,2,3} so that, for any adjacent vertices u and v, the sum of weights of edges incident to u differs from the sum of weights of edges incident to v." This survey paper presents the current state of research on the 1-2-3 Conjecture and the many variants that have been proposed in its short but active history.Comment: 30 pages, 2 tables, submitted for publicatio

Enumeration of graphs with given weighted number of connected components

We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of bipartite graphs of given order, size and number of connected components

On Chromatic Zagreb Indices of Certain Graphs

In this paper we introduce a variation of the well-known Zagreb indices by considering a proper vertex colouring of a graph $G$. The chromatic Zagreb indices are defined in terms of the parameter $c(v), v \in V(G)$ instead of the invariant $d_G(v)$. The notion of chromatically stable graphs is also introduced.Comment: 14 page

Interval edge-colorings of composition of graphs

An edge-coloring of a graph $G$ with consecutive integers $c_{1},\ldots,c_{t}$ is called an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable graphs is denoted by $\mathfrak{N}$. In 2004, Giaro and Kubale showed that if $G,H\in \mathfrak{N}$, then the Cartesian product of these graphs belongs to $\mathfrak{N}$. In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the first author showed that if $G,H\in \mathfrak{N}$ and $H$ is a regular graph, then $G[H]\in \mathfrak{N}$. In this paper, we prove that if $G\in \mathfrak{N}$ and $H$ has an interval coloring of a special type, then $G[H]\in \mathfrak{N}$. Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if $G\in \mathfrak{N}$ and $T$ is a tree, then $G[T]\in \mathfrak{N}$.Comment: 12 pages, 3 figure