On a Conjecture about the Sombor Index of Graphs

Let G be a graph with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 and SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) is the degree of the vertex v in G. We denote by Hn,ν the graph constructed from the star Sn by adding ν edge(s), 0≤ν≤n−2, between a fixed pendent vertex and ν other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph Hn,ν has the maximum Sombor index among all connected ν-cyclic graphs of order n, where 0≤ν≤n−2. In some earlier works, the validity of this conjecture was proved for ν≤5. In this paper, we confirm that this conjecture is true, when ν=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n−ν−2 is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained.

On a conjecture of Laplacian energy of trees

Let [Formula: see text] be a simple graph with [Formula: see text] vertices, [Formula: see text] edges having Laplacian eigenvalues [Formula: see text]. The Laplacian energy LE[Formula: see text] is defined as LE[Formula: see text], where [Formula: see text] is the average degree of [Formula: see text]. Radenković and Gutman conjectured that among all trees of order [Formula: see text], the path graph [Formula: see text] has the smallest Laplacian energy. Let [Formula: see text] be the family of trees of order [Formula: see text] having diameter [Formula: see text]. In this paper, we show that Laplacian energy of any tree [Formula: see text] is greater than the Laplacian energy of [Formula: see text], thereby proving the conjecture for all trees of diameter [Formula: see text]. We also show the truth of conjecture for all trees with number of non-pendent vertices at most [Formula: see text]. Further, we give some sufficient conditions for the conjecture to hold for a tree of order [Formula: see text].

Study of Circular Distance in Graphs

Circular distance between vertices of a graph has a significant role, which is defined as summation of detour distance and geodesic distance. Attention is paid, this is metric on the set of all vertices of graph and it plays an important role in graph theory. Some bounds have been carried out for circular distance in terms of pendent vertices of graph . Some results and properties have been found for circular distance for some classes of graphs and applied this distance to Cartesian product of graphs〖 P〗_2×C_n. Including 〖 P〗_2×C_n, some graphs acted as a circular self-centered. Using this circular distance there exists some relations between various radii and diameters in path graphs. The possible applications were briefly discussed.

Extremum Modified First Zagreb Connection Index of n-Vertex Trees with Fixed Number of Pendent Vertices

The modified first Zagreb connection index ZC1∗ is a graph invariant that appeared about fifty years ago within a study of molecular modeling, and after a long time, it has been revisited in two papers ((Ali and Trinajstić, 2018) and (Naji et al., 2017)) independently. For a graph G, this graph invariant is defined as ZC1∗G=∑v∈VGdvτv, where dv is the degree of the vertex v and τv is the connection number of v (that is, the number of vertices having distance 2 from v). In this paper, the graphs with maximum/minimum ZC1∗ value are characterized from the class of all n-vertex trees with fixed number of pendent vertices (that are the vertices of degree 1).

On the maximum ABC index of bipartite graphs without pendent vertices

For a simple graph G, the atom–bond connectivity index (ABC) of G is defined as ABC(G) = $\sum_{uv\in{}E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}},$where d(v) denotes the degree of vertex v of G. In this paper, we prove that for any bipartite graph G of order n ≥ 6, size 2n − 3 with δ(G) ≥ 2, $ABC(G)\leq{}\sqrt{2}(n-6)+2\sqrt{\frac{3(n-2)}{n-3}}+2,$and we characterize all extreme bipartite graphs.

The maximal \alpha-index of trees with k pendent vertices and its computation

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. The $\alpha-$ index of $G$ is the spectral radius $\rho_{\alpha}\left( G\right)$ of the matrix $A_{\alpha}\left( G\right)=\alpha D\left( G\right) +(1-\alpha)A\left( G\right)$ where $\alpha \in [0,1]$. Let $T_{n,k}$ be the tree of order $n$ and $k$ pendent vertices obtained from a star $K_{1,k}$ and $k$ pendent paths of almost equal lengths attached to different pendent vertices of $K_{1,k}$. It is shown that if $\alpha\in\left[ 0,1\right) $ and $T$ is a tree of order $n$ with $k$ pendent vertices then% \[ \rho_{\alpha}(T)\leq\rho_{\alpha}(T_{n,k}), \] with equality holding if and only if $T=T_{n,k}$. This result generalizes a theorem of Wu, Xiao and Hong \cite{WXH05} in which the result is proved for the adjacency matrix ($\alpha=0$). Let $q=[\frac{n-1}{k}]$ and $n-1=kq+r$, $0 \leq r \leq k-1$. It is also obtained that the spectrum of $A_{\alpha}(T_{n,k})$ is the union of the spectra of two special symmetric tridiagonal matrices of order $q$ and $q+1$ when $r=0$ or the union of the spectra of three special symmetric tridiagonal matrices of order $q$, $q+1$ and $2q+2$ when $r \neq 0$. Thus the $\alpha-$ index of $T_{n,k}$ can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order $q+1$ if $r=0$ or order $2q+2$ if $r\neq 0$.

Terminal Wiener Index of Balanced Trees

for A Tree T, The Terminal Wiener Index TW(T) Is Defined As Half The Sum Of All Distances Of The Form D(U, V), Where The Summation Is Over All Possible Pairs Of Pendent Vertices U,V In T. WeConsiderAClass Of Balanced Binary Trees Called 1-Trees And Compute Their Terminal Wiener Index Values. We Also Determine 1-Trees With Minimum And Maximum Terminal Wiener Index