Graphs and Interconnection networks: diameter and vulnerability

NEW

Problem: Reconstructed statement (standard degree/diameter reading): for every integer defect $\sigma\ge 0$, does there exist a finite simple undirected graph $G$ with diameter exactly $2$, maximum degree exactly $\Delta$ for some integer $\Delta$, and

The prompt’s “degree $\Delta$” is interpreted as maximum degree $\Delta$, as supported by the degree/diameter context and the rerank note. Literally allowing nonintegral $\sigma$ is impossible, since $|V(G)|$ and $\Delta$ are integers; the natural repair is $\sigma\in\mathbb Z_{\ge0}$.

Result: The repaired statement is true.

Use the following blow-up fact. If $H$ has diameter $2$ and positive integer weights $w_v$, replace each $v\in V(H)$ by an independent set of size $w_v$, joining two parts completely when the corresponding vertices of $H$ are adjacent. The resulting graph has diameter $2$, order $\sum_v w_v$, and maximum degree

Two interval constructions:

1. For every $\Delta\ge2$ and every $n\in[\Delta+1,2\Delta]$, $K_{\Delta,n-\Delta}$ has diameter $2$, maximum degree $\Delta$, and order $n$. Thus it realizes all defects

2. Let $P$ be the Petersen graph, viewed as $KG(5,2)$. Fix $v$, let $A=N(v)$, and write $\Delta=3q+r$, $0\le r<3$. Give the three vertices of $A$ weights $q,q,q+r$, and give the other seven vertices arbitrary weights in $[1,q]$. Then $v$’s neighbor-weight sum is $\Delta$, and every other neighbor-weight sum is at most $(q+r)+2q=\Delta$. Hence every order

is realized. So the realized defects include

For $\Delta\ge6$, these Petersen intervals and complete-bipartite intervals combine into

Moreover $C_6=[17,30]$, and

for all $\Delta\ge6$. Thus the $C_\Delta$ cover every integer defect $\sigma\ge17$.

It remains only to realize $\sigma<17$. The above constructions already cover

The missing values are handled by explicit $C_5$-blow-ups and one circulant graph:

Each has diameter $2$, order $\Delta^2+1-\sigma$, and maximum degree $\Delta$.

For $\sigma=13$, take the circulant graph on $\mathbb Z_{13}$ with generators $\{\pm1,\pm5\}$, which has degree $4$ and diameter $2$, and add the edge $\{0,2\}$. The resulting graph has maximum degree $5$, order $13$, diameter $2$, and

Thus every integer $\sigma\ge0$ is realized.

Audit: The proof uses finite simple graphs, maximum degree exactly $\Delta$, and diameter exactly $2$. $\Delta$ is existential and may depend on $\sigma$. No regularity assumption is introduced.

Citation: No external resolution is used; the answer is by explicit construction.