Abstract The transmission of a vertex u in a connected graph G is defined as σ(u) = Σ v∈V(G) d(u, v) and reciprocal transmission of a vertex u is defined as rs(u)=∑v∈V(G)1d(u,v) rs(u) = \sum\nolimits_{v \in V\left( G \right)} {{1 \over {d\left( {u,v} \right)}}} , where d(u, v) is the distance between vertex u and v in G. In this paper we define new distance based topological index of a connected graph G called transmission-reciprocal transmission index TRT(G)=∑uv∈E(G)(σ(u)rs(u)+σ(v)rs(v)) TRT\left( G \right) = \sum\nolimits_{uv \in E\left( G \right)} {\left( {{{\sigma \left( u \right)} \over {rs\left( u \right)}} + {{\sigma \left( v \right)} \over {rs\left( v \right)}}} \right)} and its coindex TRT¯(G)=∑uv∉E(G)(σ(u)rs(u)+σ(v)rs(v)) \overline {TRT} \left( G \right) = \sum\nolimits_{uv \notin E\left( G \right)} {\left( {{{\sigma \left( u \right)} \over {rs\left( u \right)}} + {{\sigma \left( v \right)} \over {rs\left( v \right)}}} \right)} , where E(G) is the edge set of a graph G and establish the relation between TRT(G) and TRT¯(G) \overline {TRT} \left( G \right) (G). Further compute this index for some standard class of graphs and obtain bounds for it.
Abstract Let A and S be the adjacency and the Seidel matrix of a graph G respectively. A-energy is the ordinary energy E(G) of a graph G defined as the sum of the absolute values of eigenvalues of A. Analogously, S-energy is the Seidel energy E S (G) of a graph G defined to be the sum of the absolute values of eigenvalues of the Seidel matrix S. In this article, certain class of A-equienergetic and S-equienergetic graphs are presented. Also some linear relations on A-energies and S-energies are given.
The eccentricity of a vertex [Formula: see text] in a graph [Formula: see text] is the maximum distance between [Formula: see text] and any other vertex of [Formula: see text] A vertex with maximum eccentricity is called a peripheral vertex. In this paper, we study the distance signless Laplacian matrix of a connected graph [Formula: see text] with respect to peripheral vertices and define the peripheral distance signless Laplacian matrix of a graph [Formula: see text], denoted by [Formula: see text]. We then give some bounds on various eigenvalues of [Formula: see text] Moreover, we define energy in terms of [Formula: see text] and give some bounds on the energy.
The number of distinct eigenvalues of the adjacency matrix of graph G is bounded below by d(G)+1, where d is the diameter of the graph. Graphs attaining this lower bound are known as minimal graphs. The spectrum of graph G, where G is a simple and undirected graph is the collection of different eigenvalues of the adjacency matrix with their multiplicities. This paper deals with the construction of non-regular minimal graphs, together with the study of their characteristic polynomial and spectra.
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