Edge transmission irregular graphs

Theory & Applications of Graphs Theory & Applications of Graphs

Betweenness centers of 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 and their localization in graph blocks, presenting sufficient conditions for graphs (in terms of diameter or block sizes) to have those centers contained in a single block. Further, we show that each graph occurs as the subgraph induced by the betweenness center of some graph (as well as the subgraph induced by the betweenness periphery). For trees, we show, by an alternative proof, that their betweenness center is always contained in a path; in addition, we enumerate trees of order at most 20 according to the order of their betweenness centers.

The Cayley Completion of a Graph

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 completion number $\gamma^{+}$ (additions only) and the Cayley edit distance $\gamma_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $\gamma^{+}(G)\ge n\Delta^{*}/(2m)-1$, where $\Delta^{*}$ is the least $d\ge\Delta$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $\gamma^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $\gamma^{+}$, while $\gamma_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $\gamma^{+}(P_n)=\gamma^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $\gamma_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.

Degree sum conditions for a graph to have bounded conflict-free connection number

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 $G$ conflict-free connected is referred to as the \emph{conflict-free connection number}. In this paper, we introduce several sharp conditions on the minimum degree sum of any $4$ independent vertices in $G$ to ensure that the conflict-free connection number of $G$ is bounded.

Unimodular Bicyclic Graphs

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 been completely characterized. In this paper, we investigate the corresponding problem for bicyclic graphs. We provide a complete characterization of unimodular bicyclic graphs and determine all possible values of $\det A(G)$ for a bicyclic graph $G$. Our study is motivated by the central role of unimodular graphs in the theory of graph inverses and their connections with eigenvalue reciprocity and other spectral properties of graphs.

Extremal Graphs for the Energy-Independence Number Inequality

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 every connected component of $G$ is an isolated vertex, a balanced complete multipartite graph, or a graph obtained by taking the disjoint union of $K_{a,\ldots,a}$ and $K_{b,\ldots,b}$, with the same number $r\ge3$ of parts, and then completely joining corresponding parts.