Vertex Transmissions Research Articles

Universal Distance Spectra of Join of Graphs

Consider G a simple connected graph. In this paper, we introduce the Universal distance matrix UD (G). For α, β, γ, δ ∈ R and β ̸= 0, the universal distance matrix UD (G) is defined asUD (G) = αTr (G) + βD(G) + γJ + δI,where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D(G) is the distance matrix of G. Here J is the all-ones matrix, and I is the identity matrix. In this paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary graph H. As a consequence, we obtain the eigenvalues of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, distance Seidal matrix and distance matrices of complementary graphs.

Applications on color (distance) signless laplacian energy of annihilator monic prime graph of commutative rings

In this study, we define the structure formation of the annihilator monic prime graph of commutative rings, whose distinct vertices X and J satisfies a condition annXJ≠annX⋃ann(J), graph is denoted by AMPG(Zn[x]/〈fx〉). The chromatic number of the annihilator monic prime graph is determined to establish the color (signless laplacian) energy of the graph. Also, we introduce color based energy of the annihilator monic prime graph, namely color distance signless Laplacian energy. Diag(Tr)+D(G)+Aχ, where Diag(Tr) denotes the diagonal matrix of the vertex transmissions in the graph, D(G) is the distance matrix of vertices whose diagonals are zero, the entries of the matrix Aχ are 1 if ηi and ηj are adjacent, −1 if ηi and ηj are non-adjacent with different colors, otherwise 0. The eigenvalues of color distance signless Laplacian matrix will be denoted by ∂1χ+≤∂2χ+≤∂3χ+≤⋯≤∂nχ+, then DSLχ+ of annihilator monic prime graph are define as EDSLχ+Znx〈fx〉=∑1n′∂iχ+-tG where t(G) is the transmission of the graph. Based on the eigenvalues, we can study some more properties of graphs. Applications of color based energy are also discuss in this paper.

On the spread of the distance signless Laplacian matrix of a graph

Abstract Let G be a connected graph with n vertices, m edges. The distance signless Laplacian matrix DQ(G) is defined as DQ(G) = Diag(Tr(G)) + D(G), where Diag(Tr(G)) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The distance signless Laplacian eigenvalues of G are the eigenvalues of DQ(G) and are denoted by δ1 Q(G), δ2 Q(G), ..., δn Q(G). δ1 Q is called the distance signless Laplacian spectral radius of DQ(G). In this paper, we obtain upper and lower bounds for SDQ (G) in terms of the Wiener index, the transmission degree and the order of the graph.

On graphs with distance Laplacian eigenvalues of multiplicity n−4

Let G be a connected simple graph with n vertices. The distance Laplacian matrix D L ( G ) is defined as D L ( G ) = Diag ( T r ) − D ( G ) , where Diag ( T r ) is the diagonal matrix of vertex transmissions and D ( G ) is the distance matrix of G. The eigenvalues of D L ( G ) are the distance Laplacian eigenvalues of G and are denoted by ∂ 1 L ( G ) ≥ ∂ 2 L ( G ) ≥ ⋯ ≥ ∂ n L ( G ) . The largest eigenvalue ∂ 1 L ( G ) is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018), and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity n − 3 . In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity n − 4 together with one of the distance Laplacian eigenvalues as n of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue n is of multiplicity n − 4 .

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

Let G be a connected graph of even order n. An odd [1, b]-factor of G is a spanning subgraph F of G such that dF(v) ∈ {1, 3, 5, ⋯, b} for any v ∈ V(G), where b is positive odd integer. The distance matrix Ɗ(G) of G is a symmetric real matrix with (i, j)-entry being the distance between the vertices vi and vj. The distance signless Laplacian matrix Q(G) of G is defined by Q(G), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η1(G) of Q(G) is called the distance signless Laplacian spectral radius of G. In this paper, we verify sharp upper bounds on the distance signless Laplacian spectral radius to guarantee the existence of an odd [1, b]-factor in a graph; we provide some graphs to show that the bounds are optimal.

Some upper and lower bounds for $D_{\alpha}$-energy of graphs

The generalized distance matrix of a connected graph $G$, denoted by $D_{\alpha}(G)$, is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G), ~~~~ 0\leq \alpha\leq 1$. Here, $D(G)$ is the distance matrix and $Tr(G)$ represents the vertex transmissions. Let $\partial_{1}\geq \partial_{2}\geq \cdots \geq \partial_{n}$ be the eigenvalues of $D_{\alpha}(G)$ and let $W(G)$ be the Wiener index. The generalizeddistance energy of $G$ can be defined as $E^{D_{\alpha}}(G)=\displaystyle\sum_{i=1}^{n}\left|\partial_i-\frac{2\alpha W(G)}{n}\right|$. In this paper, we develop some new theory regarding the generalized distance energy $E^{D_{\alpha}}(G)$ for a connected graph $G$. We obtain some sharp upper and lower bounds for$E^{D_{\alpha}}(G)$ connecting a wide range of parameters in graph theory including the maximum degree $\Delta$, the Wiener index $W(G)$, the diameter $d$, the transmission degrees, and the generalized distance spectral spread $D_{\alpha}S(G)$. We characterized the special graph classes that attain the bounds.

On generalized distance spectral radius and generalized distance energy of graphs

For a simple connected graph [Formula: see text], let [Formula: see text] and [Formula: see text] be the distance matrix and the diagonal matrix of the vertex transmissions, respectively. The convex linear combination [Formula: see text] of [Formula: see text] and [Formula: see text] is defined as, [Formula: see text], [Formula: see text]. The matrix [Formula: see text], known as generalized distance matrix, is effective in merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral radius and energy of the generalized distance matrix [Formula: see text] of a graph [Formula: see text]. We obtain bounds for the generalized distance spectral radius and generalized distance energy of connected graphs in terms of various parameters associated with the structure of graph.

New Bounds for the Generalized Distance Spectral Radius/Energy of Graphs

Let G be a simple connected graph with vertex set V G = v 1 , v 2 , … , v n and d v i be the degree of the vertex v i . Let D G be the distance matrix and T r G be the diagonal matrix of the vertex transmissions of G . The generalized distance matrix of G is defined as D α G = α T r G + 1 − α D G , where 0 ≤ α ≤ 1 . If λ 1 , λ 2 , … , λ n are the eigenvalues of D α G , then the generalized distance spectral radius of G is defined as ρ D α G = max 1 ≤ i ≤ n λ i . The generalized distance energy of G is E D α G = ∑ i = 1 n | λ i − 2 α W G / n | , where W G is the Wiener index of G . In this paper, we give some bounds of the generalized distance spectral radius and the generalized distance energy.

Sharp bounds for spectral radius of Dα-matrix of graphs

For a simple connected graph [Formula: see text], the generalized distance matrix [Formula: see text] is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are, respectively, the diagonal matrix of vertex transmission degrees and the distance matrix of [Formula: see text]. In this paper, we will study the spectral radius of the generalized distance matrix [Formula: see text] of a connected graph [Formula: see text]. We obtain some upper and lower bounds for the generalized distance spectral radius in terms of various parameters, like the vertex transmission degrees, the average transmission 2-degrees, etc., associated with the structure of the graph. We characterize the extremal graphs attaining these bounds. Further, we show that our bounds improve some previously known bounds existing in the literature.

Eigenvalue localization and Geršgorin disc-related problems on distance and distance-related matrices of graphs

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph [Formula: see text] are denoted by [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] is the diagonal matrix of vertex transmission. Ger[Formula: see text]gorin discs for any [Formula: see text] square matrix [Formula: see text] are the discs [Formula: see text], where [Formula: see text]. The famous Ger[Formula: see text]gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger[Formula: see text]gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger[Formula: see text]gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger[Formula: see text]gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger[Formula: see text]gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].

On the normalized distance laplacian eigenvalues of graphs

The normalized distance Laplacian matrix (DL-matrix) of a connected graph Γ is defined by DL(Γ)=I−Tr(Γ)−1/2D(Γ)Tr(Γ)−1/2, where D(Γ) is the distance matrix and Tr(Γ) is the diagonal matrix of the vertex transmissions in Γ. In this article, we present interesting spectral properties of DL(Γ)-matrix. We characterize the graphs having exactly two distinct DL-eigenvalues which in turn solves a conjecture proposed in [26]. We characterize the complete multipartite graphs with three distinct DL-eigenvalues. We present the bounds for the DL-spectral radius and the second smallest eigenvalue of DL(Γ)-matrix and identify the candidate graphs attaining them. We also identify the classes of graphs whose second smallest DL-eigenvalue is 1 and relate it with the distance spectrum of such graphs. Further, we introduce the concept of the trace norm (the normalized distance Laplacian energy DLE(Γ) of Γ) of I−DL(Γ). We obtain some bounds and characterize the corresponding extremal graphs.

On the Dα-spectral radius of cacti and bicyclic graphs

The Dα-matrix of a connected graph G is defined asDα(G)=αTr(G)+(1−α)D(G), where 0≤α<1, Tr(G) is the diagonal matrix of the vertex transmissions of G and D(G) is the distance matrix of G. The largest eigenvalue of Dα(G) is called the Dα-spectral radius of G. In this paper, we determine the unique graph with minimum Dα-spectral radius among the cacti of order n with k (k≥1) cycles and at least one pendent vertex. We also show that the Dα-spectral radius of Bn⁎ (the graph obtained from a star of order n by adding two nonadjacent edges) is less than the Dα-spectral radius of the most of bicyclic graphs of order n.

On Transmission Irregular Cubic Graphs of an Arbitrary Order

The transmission of a vertex v of a graph G is the sum of distances from v to all the other vertices of G. A transmission irregular graph (TI graph) has mutually distinct vertex transmissions. In 2018, Alizadeh and Klavžar posed the following question: do there exist infinite families of regular TI graphs? An infinite family of TI cubic graphs of order 118+72k, k≥0, was constructed by Dobrynin in 2019. In this paper, we study the problem of finding TI cubic graphs for an arbitrary number of vertices. It is shown that there exists a TI cubic graph of an arbitrary even order n≥22. Almost all constructed graphs are contained in twelve infinite families.

On spectral radius and Nordhaus-Gaddum type inequalities of the generalized distance matrix of graphs

If $Tr(G)$ and $D(G)$ are respectively the diagonal matrix of vertex transmission degrees and distance matrix of a connected graph $G$, the generalized distance matrix $D_{\alpha}(G)$ is defined as $D_{\alpha}(G)=\alpha ~Tr(G)+(1-\alpha)~D(G)$, where $0\leq \alpha \leq 1$. If $\rho_1 \geq \rho_2 \geq \dots \geq \rho_n$ are the eigenvalues of $D_{\alpha}(G)$, the largest eigenvalue $\rho_1$ (or $\rho_{\alpha}(G)$) is called the spectral radius of the generalized distance matrix $D_{\alpha}(G)$. The generalized distance energy is defined as $E^{D_{\alpha}}(G)=\sum_{i=1}^{n}\left|\rho_i -\frac{2\alpha W(G)}{n}\right|$, where $W(G)$ is the Wiener index of $G$. In this paper, we obtain the bounds for the spectral radius $\rho_{\alpha}(G)$ and the generalized distance energy of $G$ involving Wiener index. We derive the Nordhaus-Gaddum type inequalities for the spectral radius and the generalized distance energy of $G$.

A Lower Bound for the Distance Laplacian Spectral Radius of Bipartite Graphs with Given Diameter

Let G be a connected, undirected and simple graph. The distance Laplacian matrix L(G) is defined as L(G)=diag(Tr)−D(G), where D(G) denotes the distance matrix of G and diag(Tr) denotes a diagonal matrix of the vertex transmissions. Denote by ρL(G) the distance Laplacian spectral radius of G. In this paper, we determine a lower bound of the distance Laplacian spectral radius of the n-vertex bipartite graphs with diameter 4. We characterize the extremal graphs attaining this lower bound.

The generalized path matrix and energy

We define the path Laplacian matrix and the path signless Laplacian matrix of a simple connected graph G as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is the path matrix and [Formula: see text] is the diagonal matrix of the vertex transmissions. The generalized path matrix is [Formula: see text], for [Formula: see text] and [Formula: see text] are the eigenvalues of [Formula: see text]. The generalized path energy can be expressed as [Formula: see text], where [Formula: see text] is the path Wiener index of G. We give basic properties of generalized path matrix [Formula: see text]. Also, some upper and lower bounds of the generalized path energy of some graphs are studied.

On distance signless Laplacian eigenvalues of zero divisor graph of commutative rings

&lt;abstract&gt;&lt;p&gt;For a simple connected graph $ G $ of order $ n $, the distance signless Laplacian matrix is defined by $ D^{Q}(G) = D(G) + Tr(G) $, where $ D(G) $ and $ Tr(G) $ is the distance matrix and the diagonal matrix of vertex transmission degrees, respectively. The zero divisor graph $ \Gamma(R) $ of a finite commutative ring $ R $ is a simple graph, whose vertex set is the set of non-zero zero divisors of $ R $ and two vertices $ v, w \in \Gamma(R) $ are edge connected whenever $ vw = wv = 0 $. In this article, we find the $ D^{Q} $-eigenvalues of zero divisor graph of the ring $ \mathbb{Z}_{n} $ for general value $ n = {p_{1}^{l_{1}}p_{2}^{l_{2}}} $, where $ p_1 &amp;lt; p_2 $ are distinct prime numbers and $ l_{1}, l_{2} \in \mathbb{N} $. Further, we investigate the $ D^{Q} $-eigenvalues of zero divisor graphs of local rings and the rings whose associated zero divisor graphs are Hamiltonian. Also, we obtain the trace norm and the Wiener index of $ \Gamma(\mathbb{Z}_{n}) $ for some special values of $ n $.&lt;/p&gt;&lt;/abstract&gt;

On the Ger\v{s}gorin disks of distance matrices of graphs

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.

Extremal results on distance Laplacian spectral radius of graphs

The distance Laplacian spectral radius of a connected graph $G$ is the largest eigenvalue of its distance Laplacian matrix $\mathcal{L}(G)$ defined as $\mathcal{L}(G)=Tr(G)-D(G)$, where $Tr(G)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. We determine the unique trees with maximum distance Laplacian spectral radius among trees of perfect matching with given maximum degree, the unique trees with second (third, respectively) maximum distance Laplacian spectral radius, and the unique bipartite unicyclic graphs with maximum distance Laplacian spectral radius. We also determine the unique graphs with minimum distance Laplacian spectral radius among bicyclic graphs.

On the Sum of Distance Laplacian Eigenvalues of Graphs

Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $\delta_{1}, ~\delta_{1},~\dots,\delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.