The transmission of a vertex $v$ in a connected graph $G$ is the sum of distances from $v$ to all vertices in $G$. A transmission irregular (TI) graph is a connected graph in which any two distinct vertices have different transmissions. We extend the concept of transmission to edges by defining the transmission of an edge as the sum of the transmissions of its two endpoints. A connected graph can now be called edge transmission irregular (ETI) if any two distinct edges have different transmissions. We show that almost all graphs are not ETI and then investigate several related order realizability problems involving chemical ETI graphs. In particular, we prove that for every $n \ge 15$, there exists a subcubic tree of order $n$ that is both TI and ETI.
This work establishes general properties of k -total bondage and finds exact values for certain graph classes including paths, cycles, wheels, complete and complete bipartite graphs.
The betweenness centrality of a vertex $v$ in a graph $G = (V,E)$ is the sum of the relative numbers of shortest paths of $G$ that pass through $v$. The vertices of $G$ which have the maximum (resp. minimum) betweenness induce the betweenness center (resp. betweenness periphery) of $G$. We study betweenness of graphs a...
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completi...
A path in an edge-coloured graph is called \emph{conflict-free} if a colour is exclusively applied to one of its edges. A graph $G$ is considered \emph{conflict-free connected} if every pair of vertices in $V(G)$ is connected by a conflict-free path. The minimum number of colours required to render a connected graph $G...
Dinh Hanh Dang, Trung Duy Doan, P. Ha et al.· 0 citations
Let $G$ be a simple undirected graph with adjacency matrix $A(G)$. A graph $G$ is said to be \emph{unimodular} if $\det A(G)\in\{-1,1\}$. A connected graph with $m$ vertices and $m+k-1$ edges is called \emph{$k$-cyclic}; in particular, a bicyclic graph has $m$ vertices and $m+1$ edges. Unimodular unicyclic graphs have...
For a graph $G$ of order $n$, let $\mathcal E(G)$ denote its adjacency energy and let $\alpha(G)$ denote its independence number. A recent theorem of Kumar and Pragada states that $$\mathcal E(G)\ge 2\bigl(n-\alpha(G)\bigr).$$ We determine all graphs attaining equality. More precisely, equality holds if and only if eve...
S. A. Mojallal· 1 citation
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