The fan graph is determined by its signless Laplacian spectrum

summary:Given a graph $G$, if there is no nonisomorphic graph $H$ such that $G$ and $H$ have the same signless Laplacian spectra, then we say that $G$ is \hbox {$Q$-DS}. In this paper we show that every fan graph $F_n$ is \hbox {$Q$-DS}, where $F_{n}=K_{1}\vee P_{n-1}$ and $n\geq 3$

Some Extremal Graphs with Respect to Sombor Index

Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of the vertex vi in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs

Comparison Between Geometric-arithmetic Indices

The concept of geometric–arithmetic indices (GA) was introduced in the chemical graph theory very recently. In this letter we compare the geometric–arithmetic indices for chemical trees, starlike trees and general trees. Moreover, we give a conjecture for general graphs. (doi: 10.5562/cca2005

Common neighborhood energies and their relations with Zagreb index

In this paper we establish connections between common neighborhood Laplacian and common neighborhood signless Laplacian energies and the first Zagreb index of a graph $\mathcal{G}$. We introduce the concepts of CNL-hyperenergetic and CNSL-hyperenergetic graphs and showed that $\mathcal{G}$ is neither CNL-hyperenergetic nor CNSL-hyperenergetic if $\mathcal{G}$ is a complete bipartite graph. We obtain certain relations between various energies of a graph. Finally, we conclude the paper with several bounds for common neighborhood Laplacian and signless Laplacian energies of a graph.Comment: 27 page

Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk (G) of G is defined by SGutk (G) = ∑S⊆V(G),|S|=k (∏v∈S degG (v)) dG (S), in which dG (S) is the Steiner distance of S and degG (v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk . We obtain sharp upper and lower bounds of SGutk (G) + SGutk (G) and SGutk (G) · SGutk (G) for a connected graph G of order n, m edges, maximum degree ∆ and minimum degree δ

The number of spanning trees of a graph

Let G be a simple connected graph of order n, m edges, maximum degree Delta(1) and minimum degree delta. Li et al. (Appl. Math. Lett. 23: 286-290, 2010) gave an upper bound on number of spanning trees of a graph in terms of n, m, Delta(1) and delta: t(G) <= delta (2m-Delta(1)-delta-1/n-3)(n-3). The equality holds if and only if G congruent to K-1,K-n-1, G congruent to K-n, G congruent to K-1 boolean OR (K-1 boolean OR Kn-2) or G congruent to K-n - e, where e is any edge of K-n. Unfortunately, this upper bound is erroneous. In particular, we show that this upper bound is not true for complete graph K-n. In this paper we obtain some upper bounds on the number of spanning trees of graph G in terms of its structural parameters such as the number of vertices (n), the number of edges (m), maximum degree (Delta(1)), second maximum degree (Delta(2)), minimum degree (delta), independence number (alpha), clique number (omega). Moreover, we give the Nordhaus-Gaddum-type result for number of spanning trees.Faculty research Fund, Sungkyunkwan UniversityKorean Government (2013R1A1A2009341)Selçuk ÜniversitesiGlaucoma Research FoundationHong Kong Baptist Universit

On Maximal Distance Energy

Let G be a graph of order n. If the maximal connected subgraph of G has no cut vertex then it is called a block. If each block of graph G is a clique then G is called clique tree. The distance energy ED(G) of graph G is the sum of the absolute values of the eigenvalues of the distance matrix D(G). In this paper, we study the properties on the eigencomponents corresponding to the distance spectral radius of some special class of clique trees. Using this result we characterize a graph which gives the maximum distance spectral radius among all clique trees of order n with k cliques. From this result, we confirm a conjecture on the maximum distance energy, which was given in Lin et al. Linear Algebra Appl 467(2015) 29-39

A new graph based on the semi-direct product of some monoids

In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of (PM). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics